| L(s) = 1 | − 2-s + 4-s − 5-s + 2·7-s − 8-s + 10-s + 11-s + 2·13-s − 2·14-s + 16-s − 4·17-s − 20-s − 22-s − 23-s − 4·25-s − 2·26-s + 2·28-s − 29-s − 32-s + 4·34-s − 2·35-s + 4·37-s + 40-s + 6·41-s + 43-s + 44-s + 46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.755·7-s − 0.353·8-s + 0.316·10-s + 0.301·11-s + 0.554·13-s − 0.534·14-s + 1/4·16-s − 0.970·17-s − 0.223·20-s − 0.213·22-s − 0.208·23-s − 4/5·25-s − 0.392·26-s + 0.377·28-s − 0.185·29-s − 0.176·32-s + 0.685·34-s − 0.338·35-s + 0.657·37-s + 0.158·40-s + 0.937·41-s + 0.152·43-s + 0.150·44-s + 0.147·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 149454 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149454 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 19 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 16 T + p T^{2} \) | 1.67.aq |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 - 9 T + p T^{2} \) | 1.79.aj |
| 83 | \( 1 + 11 T + p T^{2} \) | 1.83.l |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 97 | \( 1 + 9 T + p T^{2} \) | 1.97.j |
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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
−13.52731206256542, −13.13321919581421, −12.51081460294537, −12.04718058411621, −11.44915631805418, −11.14677648491927, −10.94734848030849, −10.21146275156351, −9.607820934499410, −9.317514433153913, −8.618517587829452, −8.298319609030235, −7.840942391702501, −7.402956159069338, −6.739950727765812, −6.343107365863191, −5.734217551776914, −5.180745858209914, −4.364340585009282, −4.134597301046591, −3.419776093241612, −2.700280944422476, −2.041551426320734, −1.532385936148884, −0.7963393499710629, 0,
0.7963393499710629, 1.532385936148884, 2.041551426320734, 2.700280944422476, 3.419776093241612, 4.134597301046591, 4.364340585009282, 5.180745858209914, 5.734217551776914, 6.343107365863191, 6.739950727765812, 7.402956159069338, 7.840942391702501, 8.298319609030235, 8.618517587829452, 9.317514433153913, 9.607820934499410, 10.21146275156351, 10.94734848030849, 11.14677648491927, 11.44915631805418, 12.04718058411621, 12.51081460294537, 13.13321919581421, 13.52731206256542
