| 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.ad |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 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.ae |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 - T + p T^{2} \) | 1.41.ab |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 + 5 T + p T^{2} \) | 1.73.f |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−12.75171038088452, −12.39150832305972, −11.86660471628294, −11.62534253988289, −10.78362279527626, −10.54651326625835, −9.989774156543397, −9.539433775590692, −9.257851052723786, −8.853917323973017, −8.382650166558465, −7.617292280010660, −7.212730098775642, −6.555470508247429, −6.285858001766810, −5.900316336234918, −5.477828067923057, −4.691701726809561, −4.334430831682788, −3.674483316903492, −3.166697395998093, −2.618570503699908, −1.891437057014639, −1.452932131774391, −1.016493207818073, 0,
1.016493207818073, 1.452932131774391, 1.891437057014639, 2.618570503699908, 3.166697395998093, 3.674483316903492, 4.334430831682788, 4.691701726809561, 5.477828067923057, 5.900316336234918, 6.285858001766810, 6.555470508247429, 7.212730098775642, 7.617292280010660, 8.382650166558465, 8.853917323973017, 9.257851052723786, 9.539433775590692, 9.989774156543397, 10.54651326625835, 10.78362279527626, 11.62534253988289, 11.86660471628294, 12.39150832305972, 12.75171038088452
