| L(s) = 1 | + 2-s + 4-s − 4·5-s + 8-s − 3·9-s − 4·10-s − 11-s + 2·13-s + 16-s − 2·17-s − 3·18-s + 4·19-s − 4·20-s − 22-s + 6·23-s + 11·25-s + 2·26-s + 9·29-s − 6·31-s + 32-s − 2·34-s − 3·36-s − 9·37-s + 4·38-s − 4·40-s + 2·41-s + 10·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.78·5-s + 0.353·8-s − 9-s − 1.26·10-s − 0.301·11-s + 0.554·13-s + 1/4·16-s − 0.485·17-s − 0.707·18-s + 0.917·19-s − 0.894·20-s − 0.213·22-s + 1.25·23-s + 11/5·25-s + 0.392·26-s + 1.67·29-s − 1.07·31-s + 0.176·32-s − 0.342·34-s − 1/2·36-s − 1.47·37-s + 0.648·38-s − 0.632·40-s + 0.312·41-s + 1.52·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7742 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7742 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 79 | \( 1 + T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 9 T + p T^{2} \) | 1.37.j |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 7 T + p T^{2} \) | 1.53.h |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 3 T + p T^{2} \) | 1.61.ad |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 - 11 T + p T^{2} \) | 1.71.al |
| 73 | \( 1 + 13 T + p T^{2} \) | 1.73.n |
| 83 | \( 1 + T + p T^{2} \) | 1.83.b |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 + 19 T + p T^{2} \) | 1.97.t |
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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
−7.43238021072890668765203578080, −6.92743011379372280899632504168, −6.09057789117808115248188516226, −5.20863350491788148628192193875, −4.69068879520955928614585612353, −3.86029937833260774254618444431, −3.20266063439440861469857445873, −2.72788776380928479227726034707, −1.16313880815545579111524880120, 0,
1.16313880815545579111524880120, 2.72788776380928479227726034707, 3.20266063439440861469857445873, 3.86029937833260774254618444431, 4.69068879520955928614585612353, 5.20863350491788148628192193875, 6.09057789117808115248188516226, 6.92743011379372280899632504168, 7.43238021072890668765203578080
