| L(s) = 1 | + 2·3-s − 2·5-s − 7-s + 9-s + 2·11-s − 4·13-s − 4·15-s − 2·17-s − 2·21-s − 23-s − 25-s − 4·27-s + 2·29-s + 4·33-s + 2·35-s + 4·37-s − 8·39-s + 6·41-s + 2·43-s − 2·45-s − 4·47-s + 49-s − 4·51-s − 4·55-s + 2·59-s + 10·61-s − 63-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s + 0.603·11-s − 1.10·13-s − 1.03·15-s − 0.485·17-s − 0.436·21-s − 0.208·23-s − 1/5·25-s − 0.769·27-s + 0.371·29-s + 0.696·33-s + 0.338·35-s + 0.657·37-s − 1.28·39-s + 0.937·41-s + 0.304·43-s − 0.298·45-s − 0.583·47-s + 1/7·49-s − 0.560·51-s − 0.539·55-s + 0.260·59-s + 1.28·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.885208684\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.885208684\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 23 | \( 1 + T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 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
−16.37923532852989, −15.96438713316392, −15.36369963138239, −14.73022648211991, −14.51179952700678, −13.83484355344771, −13.18091371320892, −12.62576151065078, −11.92712886512202, −11.51987010635503, −10.78419960245766, −9.867451914873135, −9.495167737502033, −8.869799910792837, −8.221652631643159, −7.731852948425754, −7.151034958300878, −6.466746197880269, −5.595367798246130, −4.599893071021827, −4.035175951833552, −3.396087356107876, −2.645295984365304, −1.992925815059567, −0.5808108658888419,
0.5808108658888419, 1.992925815059567, 2.645295984365304, 3.396087356107876, 4.035175951833552, 4.599893071021827, 5.595367798246130, 6.466746197880269, 7.151034958300878, 7.731852948425754, 8.221652631643159, 8.869799910792837, 9.495167737502033, 9.867451914873135, 10.78419960245766, 11.51987010635503, 11.92712886512202, 12.62576151065078, 13.18091371320892, 13.83484355344771, 14.51179952700678, 14.73022648211991, 15.36369963138239, 15.96438713316392, 16.37923532852989
