| L(s) = 1 | − 2-s + 3-s − 4-s − 6-s + 3·8-s + 9-s − 11-s − 12-s + 13-s − 16-s − 18-s + 6·19-s + 22-s + 3·24-s − 5·25-s − 26-s + 27-s + 10·29-s − 8·31-s − 5·32-s − 33-s − 36-s + 6·37-s − 6·38-s + 39-s − 6·41-s − 2·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s + 1.06·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s + 0.277·13-s − 1/4·16-s − 0.235·18-s + 1.37·19-s + 0.213·22-s + 0.612·24-s − 25-s − 0.196·26-s + 0.192·27-s + 1.85·29-s − 1.43·31-s − 0.883·32-s − 0.174·33-s − 1/6·36-s + 0.986·37-s − 0.973·38-s + 0.160·39-s − 0.937·41-s − 0.304·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123981 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123981 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 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
−13.74929025318704, −13.50466373977498, −12.83455874522193, −12.43159932421195, −11.78885367212698, −11.29583060863006, −10.71219298127821, −10.16488200642532, −9.862329800839249, −9.279729391007614, −9.025824417127450, −8.270021140967038, −8.064416559661328, −7.516116892000429, −7.073000578099856, −6.403510296651692, −5.655095504277554, −5.208505463114476, −4.574323631173810, −4.080545204698789, −3.403953472152221, −2.945370346304655, −2.112158223609268, −1.457414865217607, −0.8740211957307015, 0,
0.8740211957307015, 1.457414865217607, 2.112158223609268, 2.945370346304655, 3.403953472152221, 4.080545204698789, 4.574323631173810, 5.208505463114476, 5.655095504277554, 6.403510296651692, 7.073000578099856, 7.516116892000429, 8.064416559661328, 8.270021140967038, 9.025824417127450, 9.279729391007614, 9.862329800839249, 10.16488200642532, 10.71219298127821, 11.29583060863006, 11.78885367212698, 12.43159932421195, 12.83455874522193, 13.50466373977498, 13.74929025318704
