| L(s) = 1 | + 5-s + 7-s − 5·11-s + 13-s − 3·17-s + 6·19-s + 7·23-s + 25-s + 6·29-s + 10·31-s + 35-s − 11·37-s − 5·41-s + 4·43-s − 2·47-s − 6·49-s + 5·53-s − 5·55-s + 61-s + 65-s + 4·67-s − 5·71-s + 6·73-s − 5·77-s − 11·79-s − 8·83-s − 3·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s − 1.50·11-s + 0.277·13-s − 0.727·17-s + 1.37·19-s + 1.45·23-s + 1/5·25-s + 1.11·29-s + 1.79·31-s + 0.169·35-s − 1.80·37-s − 0.780·41-s + 0.609·43-s − 0.291·47-s − 6/7·49-s + 0.686·53-s − 0.674·55-s + 0.128·61-s + 0.124·65-s + 0.488·67-s − 0.593·71-s + 0.702·73-s − 0.569·77-s − 1.23·79-s − 0.878·83-s − 0.325·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.266688144\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.266688144\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 7 T + p T^{2} \) | 1.23.ah |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + 11 T + p T^{2} \) | 1.37.l |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 5 T + p T^{2} \) | 1.53.af |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 + 7 T + p T^{2} \) | 1.89.h |
| 97 | \( 1 + 13 T + p T^{2} \) | 1.97.n |
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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
−7.68693676604949439671347270208, −7.02313765035128212540049865852, −6.42274847579947540880706979198, −5.40915322282046320753400186170, −5.09929736483500995034623978610, −4.42063389905168589515732204901, −3.12351094414878180406662805444, −2.78559065578662385079331693806, −1.72204133107245569291955922799, −0.73378386565205171274688099844,
0.73378386565205171274688099844, 1.72204133107245569291955922799, 2.78559065578662385079331693806, 3.12351094414878180406662805444, 4.42063389905168589515732204901, 5.09929736483500995034623978610, 5.40915322282046320753400186170, 6.42274847579947540880706979198, 7.02313765035128212540049865852, 7.68693676604949439671347270208
