| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s − 7-s + 8-s + 9-s − 10-s − 12-s + 6·13-s − 14-s + 15-s + 16-s − 6·17-s + 18-s − 20-s + 21-s − 4·23-s − 24-s + 25-s + 6·26-s − 27-s − 28-s − 2·29-s + 30-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s + 1.66·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.223·20-s + 0.218·21-s − 0.834·23-s − 0.204·24-s + 1/5·25-s + 1.17·26-s − 0.192·27-s − 0.188·28-s − 0.371·29-s + 0.182·30-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−15.65453482807119, −15.34189190048782, −14.48236705504149, −13.83836920455973, −13.50525240653707, −12.88627008601765, −12.48510895613173, −11.84005039389615, −11.28276186408103, −10.89741772701392, −10.49607596244877, −9.672103426161278, −8.884625221143263, −8.547768641446548, −7.660589302819731, −7.120776090715242, −6.482287632969336, −5.943277450427470, −5.618636020098210, −4.536953053888091, −4.179361287684056, −3.640773004794497, −2.806316947982522, −1.959244751831460, −1.067508753955517, 0,
1.067508753955517, 1.959244751831460, 2.806316947982522, 3.640773004794497, 4.179361287684056, 4.536953053888091, 5.618636020098210, 5.943277450427470, 6.482287632969336, 7.120776090715242, 7.660589302819731, 8.547768641446548, 8.884625221143263, 9.672103426161278, 10.49607596244877, 10.89741772701392, 11.28276186408103, 11.84005039389615, 12.48510895613173, 12.88627008601765, 13.50525240653707, 13.83836920455973, 14.48236705504149, 15.34189190048782, 15.65453482807119
