| L(s) = 1 | + 2-s + 4-s − 4·5-s − 7-s + 8-s − 4·10-s − 6·11-s − 14-s + 16-s + 17-s − 4·20-s − 6·22-s + 4·23-s + 11·25-s − 28-s − 8·29-s + 32-s + 34-s + 4·35-s − 4·37-s − 4·40-s − 2·41-s − 8·43-s − 6·44-s + 4·46-s − 8·47-s + 49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.78·5-s − 0.377·7-s + 0.353·8-s − 1.26·10-s − 1.80·11-s − 0.267·14-s + 1/4·16-s + 0.242·17-s − 0.894·20-s − 1.27·22-s + 0.834·23-s + 11/5·25-s − 0.188·28-s − 1.48·29-s + 0.176·32-s + 0.171·34-s + 0.676·35-s − 0.657·37-s − 0.632·40-s − 0.312·41-s − 1.21·43-s − 0.904·44-s + 0.589·46-s − 1.16·47-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361998 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361998 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 16 T + p T^{2} \) | 1.67.aq |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.85480902830576, −12.62902967098831, −12.09478084122058, −11.66693766844253, −11.14179724451683, −10.98244732313886, −10.36412396242332, −10.01541700025167, −9.325702638624573, −8.665110850522089, −8.290420868241052, −7.822548344730858, −7.448809815559810, −7.090927445751235, −6.604202570451931, −5.900501264878698, −5.301674253020800, −4.954812902577582, −4.608691214210252, −3.776746526080558, −3.543973862087281, −3.070517624496225, −2.581246511097683, −1.861336569078939, −1.001697977196357, 0, 0,
1.001697977196357, 1.861336569078939, 2.581246511097683, 3.070517624496225, 3.543973862087281, 3.776746526080558, 4.608691214210252, 4.954812902577582, 5.301674253020800, 5.900501264878698, 6.604202570451931, 7.090927445751235, 7.448809815559810, 7.822548344730858, 8.290420868241052, 8.665110850522089, 9.325702638624573, 10.01541700025167, 10.36412396242332, 10.98244732313886, 11.14179724451683, 11.66693766844253, 12.09478084122058, 12.62902967098831, 12.85480902830576
