| L(s) = 1 | − 3·5-s + 7-s − 6·11-s + 3·17-s + 2·19-s + 4·25-s − 29-s − 4·31-s − 3·35-s + 5·37-s − 41-s + 6·43-s + 2·47-s + 49-s − 53-s + 18·55-s + 6·59-s − 13·61-s + 12·67-s − 2·71-s + 5·73-s − 6·77-s − 10·79-s − 6·83-s − 9·85-s − 6·89-s − 6·95-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.377·7-s − 1.80·11-s + 0.727·17-s + 0.458·19-s + 4/5·25-s − 0.185·29-s − 0.718·31-s − 0.507·35-s + 0.821·37-s − 0.156·41-s + 0.914·43-s + 0.291·47-s + 1/7·49-s − 0.137·53-s + 2.42·55-s + 0.781·59-s − 1.66·61-s + 1.46·67-s − 0.237·71-s + 0.585·73-s − 0.683·77-s − 1.12·79-s − 0.658·83-s − 0.976·85-s − 0.635·89-s − 0.615·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 340704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 340704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 41 | \( 1 + T + p T^{2} \) | 1.41.b |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 - 5 T + p T^{2} \) | 1.73.af |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−12.64783698860356, −12.42998940213255, −11.85919209247345, −11.38905946273795, −11.04558063586670, −10.64577924273526, −10.15864217333773, −9.661586500437501, −9.121877069944787, −8.478915739112354, −8.082738996262712, −7.787090287222868, −7.348992194647571, −7.111045364503093, −6.185466430721688, −5.698822800544640, −5.174712109661931, −4.888852592845263, −4.116094688335912, −3.865634689478717, −3.091256614963652, −2.782486809181458, −2.132496023501723, −1.314714024934616, −0.6080748180258114, 0,
0.6080748180258114, 1.314714024934616, 2.132496023501723, 2.782486809181458, 3.091256614963652, 3.865634689478717, 4.116094688335912, 4.888852592845263, 5.174712109661931, 5.698822800544640, 6.185466430721688, 7.111045364503093, 7.348992194647571, 7.787090287222868, 8.082738996262712, 8.478915739112354, 9.121877069944787, 9.661586500437501, 10.15864217333773, 10.64577924273526, 11.04558063586670, 11.38905946273795, 11.85919209247345, 12.42998940213255, 12.64783698860356
