| L(s) = 1 | + 2·3-s + 2·7-s + 9-s + 2·11-s − 2·13-s − 17-s − 8·19-s + 4·21-s − 6·23-s − 4·27-s − 2·29-s − 6·31-s + 4·33-s + 2·37-s − 4·39-s − 6·41-s − 8·43-s + 8·47-s − 3·49-s − 2·51-s − 6·53-s − 16·57-s − 10·61-s + 2·63-s + 12·67-s − 12·69-s − 6·71-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.755·7-s + 1/3·9-s + 0.603·11-s − 0.554·13-s − 0.242·17-s − 1.83·19-s + 0.872·21-s − 1.25·23-s − 0.769·27-s − 0.371·29-s − 1.07·31-s + 0.696·33-s + 0.328·37-s − 0.640·39-s − 0.937·41-s − 1.21·43-s + 1.16·47-s − 3/7·49-s − 0.280·51-s − 0.824·53-s − 2.11·57-s − 1.28·61-s + 0.251·63-s + 1.46·67-s − 1.44·69-s − 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−7.85993318905395599558324003715, −7.04900491180823825979959054378, −6.31477029274768471345854341829, −5.47954157508681642844632097128, −4.50580756842865757559681134090, −3.98669669826528938513565883188, −3.16169665756874818643723760412, −2.07571643927954135034781609438, −1.81656707965426112316039617918, 0,
1.81656707965426112316039617918, 2.07571643927954135034781609438, 3.16169665756874818643723760412, 3.98669669826528938513565883188, 4.50580756842865757559681134090, 5.47954157508681642844632097128, 6.31477029274768471345854341829, 7.04900491180823825979959054378, 7.85993318905395599558324003715
