| L(s) = 1 | + 2-s + 4-s − 4·5-s + 2·7-s + 8-s − 4·10-s − 6·13-s + 2·14-s + 16-s + 4·19-s − 4·20-s + 6·23-s + 11·25-s − 6·26-s + 2·28-s − 4·29-s + 6·31-s + 32-s − 8·35-s + 4·37-s + 4·38-s − 4·40-s − 10·41-s − 4·43-s + 6·46-s − 4·47-s − 3·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.78·5-s + 0.755·7-s + 0.353·8-s − 1.26·10-s − 1.66·13-s + 0.534·14-s + 1/4·16-s + 0.917·19-s − 0.894·20-s + 1.25·23-s + 11/5·25-s − 1.17·26-s + 0.377·28-s − 0.742·29-s + 1.07·31-s + 0.176·32-s − 1.35·35-s + 0.657·37-s + 0.648·38-s − 0.632·40-s − 1.56·41-s − 0.609·43-s + 0.884·46-s − 0.583·47-s − 3/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5202 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5202 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 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
−7.68986028571729292185975214688, −7.27410674037349923323338334061, −6.59832993684411343534035733044, −5.25500463805859801932517887782, −4.84103496801237614321897152151, −4.29084785695435239225128527369, −3.30532624157795700885852815822, −2.75859393064716467363946571088, −1.37109721416970266775347916606, 0,
1.37109721416970266775347916606, 2.75859393064716467363946571088, 3.30532624157795700885852815822, 4.29084785695435239225128527369, 4.84103496801237614321897152151, 5.25500463805859801932517887782, 6.59832993684411343534035733044, 7.27410674037349923323338334061, 7.68986028571729292185975214688
