| L(s) = 1 | − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s + 3·11-s + 2·13-s + 14-s + 16-s − 4·19-s + 20-s − 3·22-s − 6·23-s + 25-s − 2·26-s − 28-s + 6·29-s − 7·31-s − 32-s − 35-s + 2·37-s + 4·38-s − 40-s + 6·41-s − 4·43-s + 3·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s − 0.353·8-s − 0.316·10-s + 0.904·11-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.917·19-s + 0.223·20-s − 0.639·22-s − 1.25·23-s + 1/5·25-s − 0.392·26-s − 0.188·28-s + 1.11·29-s − 1.25·31-s − 0.176·32-s − 0.169·35-s + 0.328·37-s + 0.648·38-s − 0.158·40-s + 0.937·41-s − 0.609·43-s + 0.452·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.504814938\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.504814938\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 263 | \( 1 - T \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−15.56656033830329, −14.88234276689225, −14.40064244112557, −13.94483562941694, −13.18299678675764, −12.76022496204925, −12.10521486760092, −11.58242966172294, −11.00168154636673, −10.39064259714459, −9.936317422069556, −9.362533965952070, −8.834419766330901, −8.340134613979066, −7.725210210638561, −6.881172817989102, −6.458019573628731, −6.003195338764703, −5.303805342203513, −4.280087701712474, −3.804286359283511, −2.954710664972344, −2.119150199229242, −1.517390385509651, −0.5563518087545075,
0.5563518087545075, 1.517390385509651, 2.119150199229242, 2.954710664972344, 3.804286359283511, 4.280087701712474, 5.303805342203513, 6.003195338764703, 6.458019573628731, 6.881172817989102, 7.725210210638561, 8.340134613979066, 8.834419766330901, 9.362533965952070, 9.936317422069556, 10.39064259714459, 11.00168154636673, 11.58242966172294, 12.10521486760092, 12.76022496204925, 13.18299678675764, 13.94483562941694, 14.40064244112557, 14.88234276689225, 15.56656033830329
