| L(s) = 1 | − 8·4-s + 3.50e3·7-s − 5.14e4·13-s + 6.55e4·16-s + 7.57e4·19-s − 7.30e5·25-s − 2.80e4·28-s + 7.02e5·31-s + 5.34e6·37-s + 7.05e6·43-s + 1.45e7·49-s + 4.11e5·52-s − 1.50e6·61-s − 1.57e6·64-s − 4.53e6·67-s + 1.10e8·73-s − 6.06e5·76-s + 4.59e7·79-s − 1.80e8·91-s − 2.94e8·97-s + 5.84e6·100-s + 3.32e8·103-s − 4.39e8·109-s + 2.29e8·112-s − 3.80e8·121-s − 5.62e6·124-s + 127-s + ⋯ |
| L(s) = 1 | − 0.0312·4-s + 1.45·7-s − 1.80·13-s + 16-s + 0.581·19-s − 1.87·25-s − 0.0455·28-s + 0.761·31-s + 2.84·37-s + 2.06·43-s + 2.53·49-s + 0.0563·52-s − 0.108·61-s − 0.0937·64-s − 0.225·67-s + 3.89·73-s − 0.0181·76-s + 1.18·79-s − 2.62·91-s − 3.32·97-s + 0.0584·100-s + 2.95·103-s − 3.11·109-s + 1.45·112-s − 1.77·121-s − 0.0237·124-s + 0.847·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43046721 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43046721 ^{s/2} \, \Gamma_{\C}(s+4)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(7.946310119\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.946310119\) |
| \(L(5)\) |
|
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 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 + p^{3} T^{2} - 1023 p^{6} T^{4} + p^{19} T^{6} + p^{32} T^{8} \) |
| 5 | $C_2^3$ | \( 1 + 29234 p^{2} T^{2} + 610486131 p^{4} T^{4} + 29234 p^{18} T^{6} + p^{32} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 - 250 p T - 55149 p^{2} T^{2} - 250 p^{9} T^{3} + p^{16} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 + 380283362 T^{2} + 98665705550450883 T^{4} + 380283362 p^{16} T^{6} + p^{32} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 25730 T - 153697821 T^{2} + 25730 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 8342551298 T^{2} + p^{16} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 18938 T + p^{8} T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 - 64711613182 T^{2} - \)\(19\!\cdots\!37\)\( T^{4} - 64711613182 p^{16} T^{6} + p^{32} T^{8} \) |
| 29 | $C_2^3$ | \( 1 + 788066452322 T^{2} + \)\(37\!\cdots\!63\)\( T^{4} + 788066452322 p^{16} T^{6} + p^{32} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 - 11338 p T - 758953437 p^{2} T^{2} - 11338 p^{9} T^{3} + p^{16} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 1335170 T + p^{8} T^{2} )^{4} \) |
| 41 | $C_2^3$ | \( 1 + 12452468931842 T^{2} + \)\(91\!\cdots\!23\)\( T^{4} + 12452468931842 p^{16} T^{6} + p^{32} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 3526150 T + 745533544899 T^{2} - 3526150 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 30967680304898 T^{2} + \)\(39\!\cdots\!83\)\( T^{4} + 30967680304898 p^{16} T^{6} + p^{32} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 80936075395298 T^{2} + p^{16} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 + 105562517046242 T^{2} - \)\(10\!\cdots\!77\)\( T^{4} + 105562517046242 p^{16} T^{6} + p^{32} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 753602 T - 191139397022877 T^{2} + 753602 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 2268890 T - 400919815724541 T^{2} + 2268890 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 1001758688017922 T^{2} + p^{16} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 27672770 T + p^{8} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 22980982 T - 988983276222237 T^{2} - 22980982 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 2352070843223138 T^{2} + \)\(45\!\cdots\!63\)\( T^{4} + 2352070843223138 p^{16} T^{6} + p^{32} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 2600204109557762 T^{2} + p^{16} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 147271010 T + 13851316792043139 T^{2} + 147271010 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
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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
−9.200875424654462829543551360597, −8.267995110519380986976500324262, −8.189665942224407446366116575572, −8.124201190581286465892162237412, −7.62195777862165552643249647601, −7.55306154539202547676694903213, −7.32155137525356874015971309843, −6.71115482857305649068374291986, −6.46355422506733528401487359863, −5.99810159198207773931778732918, −5.56741581034135348180565652314, −5.33127829980017769955031612550, −5.32021233636220806539590870002, −4.51073960504125088675098730060, −4.39399955133605347117491956910, −4.15667044861523077177629295744, −3.72642963574617506583762016779, −3.03518218341646688378629520754, −2.77602423495116183537920251159, −2.21092675827630173278218333157, −2.14907356405108403945777559391, −1.62248091786135455156413435329, −0.926557038514433326816351671310, −0.76263154303218937539059003787, −0.45842670445318595436787465684,
0.45842670445318595436787465684, 0.76263154303218937539059003787, 0.926557038514433326816351671310, 1.62248091786135455156413435329, 2.14907356405108403945777559391, 2.21092675827630173278218333157, 2.77602423495116183537920251159, 3.03518218341646688378629520754, 3.72642963574617506583762016779, 4.15667044861523077177629295744, 4.39399955133605347117491956910, 4.51073960504125088675098730060, 5.32021233636220806539590870002, 5.33127829980017769955031612550, 5.56741581034135348180565652314, 5.99810159198207773931778732918, 6.46355422506733528401487359863, 6.71115482857305649068374291986, 7.32155137525356874015971309843, 7.55306154539202547676694903213, 7.62195777862165552643249647601, 8.124201190581286465892162237412, 8.189665942224407446366116575572, 8.267995110519380986976500324262, 9.200875424654462829543551360597
