| L(s) = 1 | + 2-s + 4-s − 5-s + 2·7-s + 8-s − 10-s + 5·13-s + 2·14-s + 16-s + 17-s − 19-s − 20-s − 6·23-s + 25-s + 5·26-s + 2·28-s + 9·29-s − 31-s + 32-s + 34-s − 2·35-s − 4·37-s − 38-s − 40-s + 6·41-s + 2·43-s − 6·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.755·7-s + 0.353·8-s − 0.316·10-s + 1.38·13-s + 0.534·14-s + 1/4·16-s + 0.242·17-s − 0.229·19-s − 0.223·20-s − 1.25·23-s + 1/5·25-s + 0.980·26-s + 0.377·28-s + 1.67·29-s − 0.179·31-s + 0.176·32-s + 0.171·34-s − 0.338·35-s − 0.657·37-s − 0.162·38-s − 0.158·40-s + 0.937·41-s + 0.304·43-s − 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.862090690\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.862090690\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 17 | \( 1 - T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−9.433299876561266540798922109824, −8.330379255828883631697848182662, −8.026077504275661231293872666033, −6.90821029334418725854960093133, −6.11394596643905626071679261434, −5.29109370401463266277531097982, −4.28236517276064695505966277810, −3.69887280245863187443405432019, −2.46037473404806064640866240691, −1.19089440743210782925474912944,
1.19089440743210782925474912944, 2.46037473404806064640866240691, 3.69887280245863187443405432019, 4.28236517276064695505966277810, 5.29109370401463266277531097982, 6.11394596643905626071679261434, 6.90821029334418725854960093133, 8.026077504275661231293872666033, 8.330379255828883631697848182662, 9.433299876561266540798922109824
