| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 7-s + 8-s + 9-s − 10-s − 12-s − 2·13-s + 14-s + 15-s + 16-s − 8·17-s + 18-s + 8·19-s − 20-s − 21-s − 6·23-s − 24-s + 25-s − 2·26-s − 27-s + 28-s + 2·29-s + 30-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 − 0.554·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 1.94·17-s + 0.235·18-s + 1.83·19-s − 0.223·20-s − 0.218·21-s − 1.25·23-s − 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.188·28-s + 0.371·29-s + 0.182·30-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 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 8 T + p T^{2} \) | 1.17.i |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 8 T + p T^{2} \) | 1.53.ai |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−15.63582965784988, −15.09834772315309, −14.59013901044862, −13.87391736392243, −13.54521221536179, −12.95812090520885, −12.23155224271387, −11.84432429950181, −11.48679245557062, −10.90468940832192, −10.37322253731907, −9.651071435681733, −9.125568285532565, −8.234601222384262, −7.782771017122966, −7.043579841184724, −6.711589229692392, −5.928643392775678, −5.261762344474296, −4.842751076844666, −4.138922404144830, −3.665605369048777, −2.646723768927480, −2.070388658481342, −1.058560554453210, 0,
1.058560554453210, 2.070388658481342, 2.646723768927480, 3.665605369048777, 4.138922404144830, 4.842751076844666, 5.261762344474296, 5.928643392775678, 6.711589229692392, 7.043579841184724, 7.782771017122966, 8.234601222384262, 9.125568285532565, 9.651071435681733, 10.37322253731907, 10.90468940832192, 11.48679245557062, 11.84432429950181, 12.23155224271387, 12.95812090520885, 13.54521221536179, 13.87391736392243, 14.59013901044862, 15.09834772315309, 15.63582965784988
