| L(s) = 1 | − 3·3-s + 7-s + 6·9-s + 3·11-s + 13-s + 5·17-s − 8·19-s − 3·21-s + 2·23-s − 9·27-s + 29-s + 2·31-s − 9·33-s + 10·37-s − 3·39-s − 6·41-s + 4·43-s + 11·47-s + 49-s − 15·51-s + 6·53-s + 24·57-s − 10·59-s + 6·63-s + 10·67-s − 6·69-s + 10·73-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.377·7-s + 2·9-s + 0.904·11-s + 0.277·13-s + 1.21·17-s − 1.83·19-s − 0.654·21-s + 0.417·23-s − 1.73·27-s + 0.185·29-s + 0.359·31-s − 1.56·33-s + 1.64·37-s − 0.480·39-s − 0.937·41-s + 0.609·43-s + 1.60·47-s + 1/7·49-s − 2.10·51-s + 0.824·53-s + 3.17·57-s − 1.30·59-s + 0.755·63-s + 1.22·67-s − 0.722·69-s + 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.339864652\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.339864652\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| good | 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 11 T + p T^{2} \) | 1.47.al |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 7 T + p T^{2} \) | 1.79.ah |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 97 | \( 1 + 3 T + p T^{2} \) | 1.97.d |
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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
−16.80903580110281, −16.09775224148322, −15.36826584528663, −14.91388182175274, −14.24389506943828, −13.54242990842736, −12.71094352662284, −12.39619487714011, −11.84327900645674, −11.25917625865576, −10.80613519933634, −10.29259579412336, −9.632309705833685, −8.867780842272940, −8.126293851630532, −7.364756108702698, −6.662418983952134, −6.159322380803484, −5.691677809557447, −4.883740568347707, −4.314132028837888, −3.687201450844578, −2.362976212785033, −1.312895210405111, −0.6739201988734905,
0.6739201988734905, 1.312895210405111, 2.362976212785033, 3.687201450844578, 4.314132028837888, 4.883740568347707, 5.691677809557447, 6.159322380803484, 6.662418983952134, 7.364756108702698, 8.126293851630532, 8.867780842272940, 9.632309705833685, 10.29259579412336, 10.80613519933634, 11.25917625865576, 11.84327900645674, 12.39619487714011, 12.71094352662284, 13.54242990842736, 14.24389506943828, 14.91388182175274, 15.36826584528663, 16.09775224148322, 16.80903580110281
