| L(s) = 1 | − 2·2-s − 4-s + 20·5-s + 16·7-s − 4·8-s − 40·10-s − 80·13-s − 32·14-s − 19·16-s − 164·17-s + 36·19-s − 20·20-s + 172·23-s + 62·25-s + 160·26-s − 16·28-s − 108·29-s − 448·31-s + 202·32-s + 328·34-s + 320·35-s + 108·37-s − 72·38-s − 80·40-s − 212·41-s − 156·43-s − 344·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/8·4-s + 1.78·5-s + 0.863·7-s − 0.176·8-s − 1.26·10-s − 1.70·13-s − 0.610·14-s − 0.296·16-s − 2.33·17-s + 0.434·19-s − 0.223·20-s + 1.55·23-s + 0.495·25-s + 1.20·26-s − 0.107·28-s − 0.691·29-s − 2.59·31-s + 1.11·32-s + 1.65·34-s + 1.54·35-s + 0.479·37-s − 0.307·38-s − 0.316·40-s − 0.807·41-s − 0.553·43-s − 1.10·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 5 T^{2} + p^{4} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 4 p T + 338 T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 16 T + 450 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 80 T + 4542 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 164 T + 16118 T^{2} + 164 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 36 T + 3242 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 172 T + 30758 T^{2} - 172 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 108 T + 14062 T^{2} + 108 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 448 T + 101646 T^{2} + 448 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 108 T + 103022 T^{2} - 108 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 212 T + 148886 T^{2} + 212 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 156 T + 63530 T^{2} + 156 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 20 T + 202454 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 132 T - 117518 T^{2} + 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 688 T + 404246 T^{2} - 688 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 96 T + 402398 T^{2} - 96 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 448 T + 455094 T^{2} - 448 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 132 T - 187322 T^{2} - 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 428 T + 808278 T^{2} + 428 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 424 T + 1031010 T^{2} + 424 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 720 T + 1262374 T^{2} + 720 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 1056 T + 1317010 T^{2} - 1056 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 52 T - 413466 T^{2} - 52 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.200256171156813607491634377249, −9.108200864758017136180630682083, −8.551703543901820107952098533409, −8.353236075466875229324673528332, −7.42680149889452129964965082216, −7.25700305430121329472647934702, −6.93005101586612902007507441807, −6.29464201062967146357037927359, −5.94615012277027016980684116841, −5.31943888165764134976885020938, −4.88833896532552044477943635161, −4.87989843017859069359208602434, −4.05046281158740398209639229582, −3.35532440775929443532092333843, −2.46305858758904788932847487478, −2.22768309200321555869336543079, −1.88853440202497061301495973394, −1.20104607779596334488881420150, 0, 0,
1.20104607779596334488881420150, 1.88853440202497061301495973394, 2.22768309200321555869336543079, 2.46305858758904788932847487478, 3.35532440775929443532092333843, 4.05046281158740398209639229582, 4.87989843017859069359208602434, 4.88833896532552044477943635161, 5.31943888165764134976885020938, 5.94615012277027016980684116841, 6.29464201062967146357037927359, 6.93005101586612902007507441807, 7.25700305430121329472647934702, 7.42680149889452129964965082216, 8.353236075466875229324673528332, 8.551703543901820107952098533409, 9.108200864758017136180630682083, 9.200256171156813607491634377249
