| L(s) = 1 | + 2·2-s − 5·4-s + 16·5-s − 28·7-s − 12·8-s + 32·10-s − 4·11-s − 32·13-s − 56·14-s − 11·16-s − 138·17-s + 180·19-s − 80·20-s − 8·22-s + 116·23-s − 40·25-s − 64·26-s + 140·28-s − 304·29-s + 62·31-s − 122·32-s − 276·34-s − 448·35-s − 340·37-s + 360·38-s − 192·40-s + 268·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 5/8·4-s + 1.43·5-s − 1.51·7-s − 0.530·8-s + 1.01·10-s − 0.109·11-s − 0.682·13-s − 1.06·14-s − 0.171·16-s − 1.96·17-s + 2.17·19-s − 0.894·20-s − 0.0775·22-s + 1.05·23-s − 0.319·25-s − 0.482·26-s + 0.944·28-s − 1.94·29-s + 0.359·31-s − 0.673·32-s − 1.39·34-s − 2.16·35-s − 1.51·37-s + 1.53·38-s − 0.758·40-s + 1.02·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700569 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700569 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 31 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - p T + 9 T^{2} - p^{4} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 16 T + 296 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 4 p T + 544 T^{2} + 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 1208 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 32 T + 4522 T^{2} + 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 138 T + 10889 T^{2} + 138 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 180 T + 21786 T^{2} - 180 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 116 T + 27248 T^{2} - 116 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 304 T + 62082 T^{2} + 304 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 340 T + 104214 T^{2} + 340 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 268 T + 79748 T^{2} - 268 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 170 T + 52951 T^{2} + 170 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 10 p T + 257463 T^{2} + 10 p^{4} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 750 T + 338027 T^{2} - 750 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 798 T + 568277 T^{2} - 798 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 216 T + 464274 T^{2} - 216 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 1004 T + 684168 T^{2} + 1004 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 286 T + 724719 T^{2} + 286 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 568 T + 639568 T^{2} + 568 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2 p T + 581901 T^{2} - 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 864 T + 1189748 T^{2} + 864 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 974 T + 1425329 T^{2} - 974 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 338 T + 469459 T^{2} + 338 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.413462437069017791935963486802, −9.398105552249257679762552650781, −8.992335829942551955202711287118, −8.622972073075243637479086468525, −7.78155235759584248175449638294, −7.14966229121515360371938466325, −7.02872215766456502023969239784, −6.53703631952639531360246444810, −5.86943804558161835346756403235, −5.65336543949013963251811259852, −5.06896077883505994566764952006, −4.96721848422479411924266298565, −3.94368160278800898236485833813, −3.83601122952310530858183001883, −3.04969192217689440785226234089, −2.58820874666618505734218482739, −2.00325464788180145627237190078, −1.27666260873600879637069724863, 0, 0,
1.27666260873600879637069724863, 2.00325464788180145627237190078, 2.58820874666618505734218482739, 3.04969192217689440785226234089, 3.83601122952310530858183001883, 3.94368160278800898236485833813, 4.96721848422479411924266298565, 5.06896077883505994566764952006, 5.65336543949013963251811259852, 5.86943804558161835346756403235, 6.53703631952639531360246444810, 7.02872215766456502023969239784, 7.14966229121515360371938466325, 7.78155235759584248175449638294, 8.622972073075243637479086468525, 8.992335829942551955202711287118, 9.398105552249257679762552650781, 9.413462437069017791935963486802
