| L(s) = 1 | + 3-s − 2·9-s + 2·11-s + 13-s − 2·17-s − 8·19-s − 23-s − 5·27-s + 9·29-s + 4·31-s + 2·33-s − 6·37-s + 39-s − 2·41-s + 9·43-s + 4·47-s − 7·49-s − 2·51-s − 13·53-s − 8·57-s − 6·59-s − 5·61-s + 6·67-s − 69-s − 2·71-s − 8·73-s − 17·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2/3·9-s + 0.603·11-s + 0.277·13-s − 0.485·17-s − 1.83·19-s − 0.208·23-s − 0.962·27-s + 1.67·29-s + 0.718·31-s + 0.348·33-s − 0.986·37-s + 0.160·39-s − 0.312·41-s + 1.37·43-s + 0.583·47-s − 49-s − 0.280·51-s − 1.78·53-s − 1.05·57-s − 0.781·59-s − 0.640·61-s + 0.733·67-s − 0.120·69-s − 0.237·71-s − 0.936·73-s − 1.91·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 9 T + p T^{2} \) | 1.43.aj |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 13 T + p T^{2} \) | 1.53.n |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + 17 T + p T^{2} \) | 1.79.r |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−8.185985696637849996584433007200, −7.08545620104147912961940481650, −6.36628873820725714821851039362, −5.90390522998923856025048093097, −4.68502280802899793951157972967, −4.17217871456396527860274650890, −3.17504311882563719684539065162, −2.47695434199269821946601113546, −1.50495458136540948839423452661, 0,
1.50495458136540948839423452661, 2.47695434199269821946601113546, 3.17504311882563719684539065162, 4.17217871456396527860274650890, 4.68502280802899793951157972967, 5.90390522998923856025048093097, 6.36628873820725714821851039362, 7.08545620104147912961940481650, 8.185985696637849996584433007200
