| L(s) = 1 | + 2-s + 4-s − 4·5-s + 7-s + 8-s − 4·10-s − 5·11-s − 13-s + 14-s + 16-s − 6·17-s − 6·19-s − 4·20-s − 5·22-s + 5·23-s + 11·25-s − 26-s + 28-s + 5·29-s − 5·31-s + 32-s − 6·34-s − 4·35-s − 9·37-s − 6·38-s − 4·40-s − 6·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.78·5-s + 0.377·7-s + 0.353·8-s − 1.26·10-s − 1.50·11-s − 0.277·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s − 1.37·19-s − 0.894·20-s − 1.06·22-s + 1.04·23-s + 11/5·25-s − 0.196·26-s + 0.188·28-s + 0.928·29-s − 0.898·31-s + 0.176·32-s − 1.02·34-s − 0.676·35-s − 1.47·37-s − 0.973·38-s − 0.632·40-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 + 9 T + p T^{2} \) | 1.37.j |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 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
−10.58160738591835707753460857513, −8.807803396493784636380359236500, −8.175177680256452460931522884422, −7.35680029991455180786399651281, −6.61672779457864729845300517948, −5.05043340362145808941393518904, −4.56312131234610716383609439911, −3.51475991148584634937839178800, −2.38730960948744802020693228411, 0,
2.38730960948744802020693228411, 3.51475991148584634937839178800, 4.56312131234610716383609439911, 5.05043340362145808941393518904, 6.61672779457864729845300517948, 7.35680029991455180786399651281, 8.175177680256452460931522884422, 8.807803396493784636380359236500, 10.58160738591835707753460857513
