| L(s) = 1 | + 15·2-s + 75·4-s − 504·5-s + 75·8-s − 7.56e3·10-s + 1.92e3·11-s + 1.81e4·13-s − 1.13e3·16-s + 1.95e4·17-s + 3.13e4·19-s − 3.78e4·20-s + 2.88e4·22-s − 1.01e5·23-s + 8.15e3·25-s + 2.72e5·26-s − 1.94e5·29-s + 7.88e4·31-s + 8.02e4·32-s + 2.93e5·34-s + 1.28e5·37-s + 4.69e5·38-s − 3.78e4·40-s − 3.65e5·41-s + 4.49e5·43-s + 1.44e5·44-s − 1.51e6·46-s + 1.57e6·47-s + ⋯ |
| L(s) = 1 | + 1.32·2-s + 0.585·4-s − 1.80·5-s + 0.0517·8-s − 2.39·10-s + 0.434·11-s + 2.29·13-s − 0.0691·16-s + 0.966·17-s + 1.04·19-s − 1.05·20-s + 0.576·22-s − 1.73·23-s + 0.104·25-s + 3.03·26-s − 1.48·29-s + 0.475·31-s + 0.432·32-s + 1.28·34-s + 0.417·37-s + 1.38·38-s − 0.0933·40-s − 0.827·41-s + 0.862·43-s + 0.254·44-s − 2.29·46-s + 2.21·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.789416667\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.789416667\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2 \wr S_4$ | \( 1 - 15 T + 75 p T^{2} - 75 p^{4} T^{3} + 563 p^{4} T^{4} - 75 p^{11} T^{5} + 75 p^{15} T^{6} - 15 p^{21} T^{7} + p^{28} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 + 504 T + 245864 T^{2} + 16436088 p T^{3} + 945912174 p^{2} T^{4} + 16436088 p^{8} T^{5} + 245864 p^{14} T^{6} + 504 p^{21} T^{7} + p^{28} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 1920 T + 47303904 T^{2} - 170564973360 T^{3} + 1058200130947886 T^{4} - 170564973360 p^{7} T^{5} + 47303904 p^{14} T^{6} - 1920 p^{21} T^{7} + p^{28} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 - 18144 T + 274398964 T^{2} - 180541542816 p T^{3} + 21711173630808534 T^{4} - 180541542816 p^{8} T^{5} + 274398964 p^{14} T^{6} - 18144 p^{21} T^{7} + p^{28} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 1152 p T + 990517208 T^{2} - 20338104959280 T^{3} + 558927253513768686 T^{4} - 20338104959280 p^{7} T^{5} + 990517208 p^{14} T^{6} - 1152 p^{22} T^{7} + p^{28} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 31320 T + 2176546108 T^{2} - 3789132064200 p T^{3} + 2482471623526350198 T^{4} - 3789132064200 p^{8} T^{5} + 2176546108 p^{14} T^{6} - 31320 p^{21} T^{7} + p^{28} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 101160 T + 14941424016 T^{2} + 948812252401080 T^{3} + 77579578451265593342 T^{4} + 948812252401080 p^{7} T^{5} + 14941424016 p^{14} T^{6} + 101160 p^{21} T^{7} + p^{28} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 194832 T + 68169175020 T^{2} + 9181092556616112 T^{3} + \)\(17\!\cdots\!94\)\( T^{4} + 9181092556616112 p^{7} T^{5} + 68169175020 p^{14} T^{6} + 194832 p^{21} T^{7} + p^{28} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 78840 T + 22399470412 T^{2} - 6212138259968280 T^{3} + \)\(10\!\cdots\!78\)\( T^{4} - 6212138259968280 p^{7} T^{5} + 22399470412 p^{14} T^{6} - 78840 p^{21} T^{7} + p^{28} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 128640 T + 276444636844 T^{2} - 23107019753427840 T^{3} + \)\(34\!\cdots\!22\)\( T^{4} - 23107019753427840 p^{7} T^{5} + 276444636844 p^{14} T^{6} - 128640 p^{21} T^{7} + p^{28} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 365040 T + 391725770744 T^{2} + 146719702189581120 T^{3} + \)\(84\!\cdots\!06\)\( T^{4} + 146719702189581120 p^{7} T^{5} + 391725770744 p^{14} T^{6} + 365040 p^{21} T^{7} + p^{28} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 449520 T + 911801001196 T^{2} - 338923246405434480 T^{3} + \)\(35\!\cdots\!42\)\( T^{4} - 338923246405434480 p^{7} T^{5} + 911801001196 p^{14} T^{6} - 449520 p^{21} T^{7} + p^{28} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 1575792 T + 2176986331196 T^{2} - 2060788106754112176 T^{3} + \)\(17\!\cdots\!54\)\( T^{4} - 2060788106754112176 p^{7} T^{5} + 2176986331196 p^{14} T^{6} - 1575792 p^{21} T^{7} + p^{28} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 1448160 T + 2314424972460 T^{2} - 1862112362801250720 T^{3} + \)\(40\!\cdots\!46\)\( p T^{4} - 1862112362801250720 p^{7} T^{5} + 2314424972460 p^{14} T^{6} - 1448160 p^{21} T^{7} + p^{28} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 3280320 T + 6359002636268 T^{2} + 7262261770225867200 T^{3} + \)\(10\!\cdots\!18\)\( T^{4} + 7262261770225867200 p^{7} T^{5} + 6359002636268 p^{14} T^{6} + 3280320 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 606960 T + 6006471729796 T^{2} - 1606415295255348240 T^{3} + \)\(20\!\cdots\!86\)\( T^{4} - 1606415295255348240 p^{7} T^{5} + 6006471729796 p^{14} T^{6} - 606960 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 3492880 T + 16253275359964 T^{2} - 44862857987077695760 T^{3} + \)\(15\!\cdots\!42\)\( T^{4} - 44862857987077695760 p^{7} T^{5} + 16253275359964 p^{14} T^{6} - 3492880 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 984 T + 23313454396080 T^{2} + 10877264513013635016 T^{3} + \)\(26\!\cdots\!74\)\( T^{4} + 10877264513013635016 p^{7} T^{5} + 23313454396080 p^{14} T^{6} + 984 p^{21} T^{7} + p^{28} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 10981440 T + 68407824224308 T^{2} - 4516109865540002880 p T^{3} + \)\(12\!\cdots\!34\)\( T^{4} - 4516109865540002880 p^{8} T^{5} + 68407824224308 p^{14} T^{6} - 10981440 p^{21} T^{7} + p^{28} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 4654544 T + 50026680158092 T^{2} - \)\(19\!\cdots\!72\)\( T^{3} + \)\(11\!\cdots\!90\)\( T^{4} - \)\(19\!\cdots\!72\)\( p^{7} T^{5} + 50026680158092 p^{14} T^{6} - 4654544 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 8126496 T + 100452515205644 T^{2} + \)\(57\!\cdots\!12\)\( T^{3} + \)\(40\!\cdots\!74\)\( T^{4} + \)\(57\!\cdots\!12\)\( p^{7} T^{5} + 100452515205644 p^{14} T^{6} + 8126496 p^{21} T^{7} + p^{28} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 11272320 T + 66412349776184 T^{2} + \)\(10\!\cdots\!00\)\( T^{3} - \)\(21\!\cdots\!94\)\( T^{4} + \)\(10\!\cdots\!00\)\( p^{7} T^{5} + 66412349776184 p^{14} T^{6} - 11272320 p^{21} T^{7} + p^{28} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 6572448 T + 235163205768916 T^{2} - \)\(14\!\cdots\!84\)\( T^{3} + \)\(26\!\cdots\!14\)\( T^{4} - \)\(14\!\cdots\!84\)\( p^{7} T^{5} + 235163205768916 p^{14} T^{6} - 6572448 p^{21} T^{7} + p^{28} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.68779734831956834378663139837, −6.56151276608749694442149422890, −6.17138833484516274045500752675, −5.99606898354885577537687426441, −5.98802540711755536878819834467, −5.49860290117087724188617347596, −5.37719382689244194070411657014, −4.97414993628642114697251455215, −4.95244710737128545181408903300, −4.33595697687063275175193037095, −4.01508819738133229017809005404, −3.95931097625176438886871125557, −3.95775501804133269876556633482, −3.63762052529442267226931591159, −3.46604848179633742635900600880, −3.20352285941484370350973627584, −2.73661009681317652862443701902, −2.46629708229138251554307658843, −1.99429725566084855725731380782, −1.84039757478793403436641290931, −1.39778115234131214091335037091, −0.949906427565342672398092058807, −0.867378062601062192935613528250, −0.62355095797320344256106553873, −0.10188128515831940023273402482,
0.10188128515831940023273402482, 0.62355095797320344256106553873, 0.867378062601062192935613528250, 0.949906427565342672398092058807, 1.39778115234131214091335037091, 1.84039757478793403436641290931, 1.99429725566084855725731380782, 2.46629708229138251554307658843, 2.73661009681317652862443701902, 3.20352285941484370350973627584, 3.46604848179633742635900600880, 3.63762052529442267226931591159, 3.95775501804133269876556633482, 3.95931097625176438886871125557, 4.01508819738133229017809005404, 4.33595697687063275175193037095, 4.95244710737128545181408903300, 4.97414993628642114697251455215, 5.37719382689244194070411657014, 5.49860290117087724188617347596, 5.98802540711755536878819834467, 5.99606898354885577537687426441, 6.17138833484516274045500752675, 6.56151276608749694442149422890, 6.68779734831956834378663139837
