| L(s) = 1 | + 12·2-s + 80·4-s + 1.24e4·8-s + 1.03e5·9-s − 2.12e5·13-s − 7.56e5·16-s + 1.71e5·17-s + 1.24e6·18-s − 2.54e6·26-s + 3.00e7·29-s − 2.14e7·32-s + 2.05e6·34-s + 8.29e6·36-s − 1.34e8·37-s − 3.40e8·41-s + 1.62e8·49-s − 1.69e7·52-s − 1.43e9·53-s + 3.60e8·58-s + 3.41e9·61-s − 1.39e8·64-s + 1.37e7·68-s + 1.29e9·72-s + 2.98e9·73-s − 1.61e9·74-s + 3.77e9·81-s − 4.08e9·82-s + ⋯ |
| L(s) = 1 | + 3/8·2-s + 5/64·4-s + 0.380·8-s + 1.75·9-s − 0.571·13-s − 0.721·16-s + 0.120·17-s + 0.658·18-s − 0.214·26-s + 1.46·29-s − 0.640·32-s + 0.0452·34-s + 0.137·36-s − 1.93·37-s − 2.93·41-s + 0.575·49-s − 0.0446·52-s − 3.43·53-s + 0.549·58-s + 4.03·61-s − 0.130·64-s + 0.00943·68-s + 0.668·72-s + 1.44·73-s − 0.726·74-s + 1.08·81-s − 1.10·82-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(11-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+5)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(4.131328334\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.131328334\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $D_{4}$ | \( 1 - 3 p^{2} T + p^{6} T^{2} - 3 p^{12} T^{3} + p^{20} T^{4} \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2 \wr C_2$ | \( 1 - 11524 p^{2} T^{2} + 28732802 p^{5} T^{4} - 11524 p^{22} T^{6} + p^{40} T^{8} \) |
| 7 | $C_2^2 \wr C_2$ | \( 1 - 23212828 p T^{2} + 96417640416042 p^{3} T^{4} - 23212828 p^{21} T^{6} + p^{40} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 - 91776568804 T^{2} + \)\(34\!\cdots\!26\)\( T^{4} - 91776568804 p^{20} T^{6} + p^{40} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 628 p^{2} T + 153356848374 T^{2} + 628 p^{12} T^{3} + p^{20} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 - 85692 T + 4021876040134 T^{2} - 85692 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 - 9955343221924 T^{2} + \)\(46\!\cdots\!66\)\( T^{4} - 9955343221924 p^{20} T^{6} + p^{40} T^{8} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 - 93909691628356 T^{2} + \)\(51\!\cdots\!06\)\( T^{4} - 93909691628356 p^{20} T^{6} + p^{40} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 15023316 T + 701527143006646 T^{2} - 15023316 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 - 2332612996197124 T^{2} + \)\(24\!\cdots\!66\)\( T^{4} - 2332612996197124 p^{20} T^{6} + p^{40} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 67204468 T + 10189261825538454 T^{2} + 67204468 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 170090076 T + 30469377593098726 T^{2} + 170090076 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 43 | $C_2^2 \wr C_2$ | \( 1 - 67155685993274596 T^{2} + \)\(20\!\cdots\!06\)\( T^{4} - 67155685993274596 p^{20} T^{6} + p^{40} T^{8} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 - 178889386570200196 T^{2} + \)\(13\!\cdots\!26\)\( T^{4} - 178889386570200196 p^{20} T^{6} + p^{40} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 718785972 T + 436515587468254294 T^{2} + 718785972 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 93873365134043036 T^{2} - \)\(34\!\cdots\!54\)\( T^{4} + 93873365134043036 p^{20} T^{6} + p^{40} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 1706041684 T + 1939939354798747446 T^{2} - 1706041684 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 - 586380343444775716 T^{2} + \)\(45\!\cdots\!86\)\( T^{4} - 586380343444775716 p^{20} T^{6} + p^{40} T^{8} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 - 5068560656654470084 T^{2} + \)\(27\!\cdots\!86\)\( T^{4} - 5068560656654470084 p^{20} T^{6} + p^{40} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 1494255068 T + 2822234732034185574 T^{2} - 1494255068 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 - 36325713459635253124 T^{2} + \)\(50\!\cdots\!66\)\( T^{4} - 36325713459635253124 p^{20} T^{6} + p^{40} T^{8} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 - 10916459259555656356 T^{2} + \)\(15\!\cdots\!06\)\( T^{4} - 10916459259555656356 p^{20} T^{6} + p^{40} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 2637360996 T + 62016549481627943206 T^{2} - 2637360996 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 7171599748 T + \)\(15\!\cdots\!94\)\( T^{2} + 7171599748 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
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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
−8.246562059895530448108828138723, −7.975763693258720413898754929714, −7.43411778492684436207819051739, −7.24874638323091166936773732837, −6.78776646942359943893489000183, −6.71610652105180683214060472719, −6.70146149126060040978299741190, −6.27482275162987612303972984264, −5.59112506288215298844474253444, −5.22463331626180358164491788966, −5.18741457696886596125163807447, −4.69257957021805422407961142464, −4.67449887259862700011348328460, −4.17868482377589520715838353712, −3.77952348102678047482890255179, −3.53713922536001678441240596703, −3.35067512805053892269937914145, −2.64618014034315348471499284770, −2.34859143293854392398916923962, −2.17880352887108893323917184991, −1.38995706535587498741794566388, −1.37575269725546342942699816025, −1.36648002897995827432891292202, −0.38723157218140613261667495941, −0.30846438791336281143509955276,
0.30846438791336281143509955276, 0.38723157218140613261667495941, 1.36648002897995827432891292202, 1.37575269725546342942699816025, 1.38995706535587498741794566388, 2.17880352887108893323917184991, 2.34859143293854392398916923962, 2.64618014034315348471499284770, 3.35067512805053892269937914145, 3.53713922536001678441240596703, 3.77952348102678047482890255179, 4.17868482377589520715838353712, 4.67449887259862700011348328460, 4.69257957021805422407961142464, 5.18741457696886596125163807447, 5.22463331626180358164491788966, 5.59112506288215298844474253444, 6.27482275162987612303972984264, 6.70146149126060040978299741190, 6.71610652105180683214060472719, 6.78776646942359943893489000183, 7.24874638323091166936773732837, 7.43411778492684436207819051739, 7.975763693258720413898754929714, 8.246562059895530448108828138723
