| L(s) = 1 | − 2-s + 4-s − 3·5-s + 7-s − 8-s + 3·10-s + 11-s + 4·13-s − 14-s + 16-s − 3·20-s − 22-s − 8·23-s + 4·25-s − 4·26-s + 28-s + 29-s − 3·31-s − 32-s − 3·35-s + 4·37-s + 3·40-s + 4·41-s − 6·43-s + 44-s + 8·46-s + 49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.34·5-s + 0.377·7-s − 0.353·8-s + 0.948·10-s + 0.301·11-s + 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.670·20-s − 0.213·22-s − 1.66·23-s + 4/5·25-s − 0.784·26-s + 0.188·28-s + 0.185·29-s − 0.538·31-s − 0.176·32-s − 0.507·35-s + 0.657·37-s + 0.474·40-s + 0.624·41-s − 0.914·43-s + 0.150·44-s + 1.17·46-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 5 T + p T^{2} \) | 1.53.f |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 5 T + p T^{2} \) | 1.73.af |
| 79 | \( 1 + 3 T + p T^{2} \) | 1.79.d |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 9 T + p T^{2} \) | 1.97.j |
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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
−15.43470460306852, −14.73889892626893, −14.16100133135370, −13.77020020774895, −12.74617124234418, −12.58769394443429, −11.68248018760262, −11.42230090877079, −11.18605603441582, −10.33430701162992, −9.934362747162149, −9.182789093156788, −8.489533488097927, −8.308318753327252, −7.674078496662032, −7.286728151029546, −6.441516045623599, −6.058700041637309, −5.229733816309030, −4.394260066709803, −3.833626646526868, −3.459458351986310, −2.459262560027879, −1.664099222779520, −0.8694062959733924, 0,
0.8694062959733924, 1.664099222779520, 2.459262560027879, 3.459458351986310, 3.833626646526868, 4.394260066709803, 5.229733816309030, 6.058700041637309, 6.441516045623599, 7.286728151029546, 7.674078496662032, 8.308318753327252, 8.489533488097927, 9.182789093156788, 9.934362747162149, 10.33430701162992, 11.18605603441582, 11.42230090877079, 11.68248018760262, 12.58769394443429, 12.74617124234418, 13.77020020774895, 14.16100133135370, 14.73889892626893, 15.43470460306852
