| L(s) = 1 | − 2-s − 4-s + 5-s + 2·7-s + 3·8-s − 10-s − 4·13-s − 2·14-s − 16-s − 6·17-s − 20-s + 6·23-s + 25-s + 4·26-s − 2·28-s − 10·29-s + 8·31-s − 5·32-s + 6·34-s + 2·35-s − 10·37-s + 3·40-s − 6·41-s + 2·43-s − 6·46-s + 2·47-s − 3·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.447·5-s + 0.755·7-s + 1.06·8-s − 0.316·10-s − 1.10·13-s − 0.534·14-s − 1/4·16-s − 1.45·17-s − 0.223·20-s + 1.25·23-s + 1/5·25-s + 0.784·26-s − 0.377·28-s − 1.85·29-s + 1.43·31-s − 0.883·32-s + 1.02·34-s + 0.338·35-s − 1.64·37-s + 0.474·40-s − 0.937·41-s + 0.304·43-s − 0.884·46-s + 0.291·47-s − 3/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 364815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 364815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 67 | \( 1 + T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 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
−12.86849153024435, −12.23316200178625, −11.76660326324644, −11.27225102371417, −10.72451056890988, −10.57963060756523, −9.881109769176415, −9.518643389479822, −9.130649671158123, −8.687087184385547, −8.272406363572223, −7.835079746614538, −7.248218599106427, −6.777392670473550, −6.575105802832756, −5.442433984257928, −5.121478124368753, −5.048117484958727, −4.151193076671261, −3.946610671155624, −3.056084292256984, −2.308035498916586, −2.017431894902530, −1.369514678418803, −0.6661269484823768, 0,
0.6661269484823768, 1.369514678418803, 2.017431894902530, 2.308035498916586, 3.056084292256984, 3.946610671155624, 4.151193076671261, 5.048117484958727, 5.121478124368753, 5.442433984257928, 6.575105802832756, 6.777392670473550, 7.248218599106427, 7.835079746614538, 8.272406363572223, 8.687087184385547, 9.130649671158123, 9.518643389479822, 9.881109769176415, 10.57963060756523, 10.72451056890988, 11.27225102371417, 11.76660326324644, 12.23316200178625, 12.86849153024435
