| L(s) = 1 | + 2-s + 4-s + 2·5-s + 8-s + 2·10-s − 2·13-s + 16-s − 6·17-s − 4·19-s + 2·20-s − 8·23-s − 25-s − 2·26-s + 2·29-s − 31-s + 32-s − 6·34-s + 10·37-s − 4·38-s + 2·40-s − 6·41-s − 8·43-s − 8·46-s + 8·47-s − 7·49-s − 50-s − 2·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.353·8-s + 0.632·10-s − 0.554·13-s + 1/4·16-s − 1.45·17-s − 0.917·19-s + 0.447·20-s − 1.66·23-s − 1/5·25-s − 0.392·26-s + 0.371·29-s − 0.179·31-s + 0.176·32-s − 1.02·34-s + 1.64·37-s − 0.648·38-s + 0.316·40-s − 0.937·41-s − 1.21·43-s − 1.17·46-s + 1.16·47-s − 49-s − 0.141·50-s − 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67518 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67518 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.088111603\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.088111603\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| 31 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−14.18296903233057, −13.48349053374247, −13.29655252366993, −12.91689400644592, −12.17773162745189, −11.67714470422617, −11.38127058263251, −10.49715254205282, −10.21216470596385, −9.795735029500446, −9.044164543791231, −8.590935245798473, −7.975248492582366, −7.327999363030190, −6.730537469121942, −6.122202465482822, −5.995127233933316, −5.156125631289420, −4.576154798164534, −4.164015334703001, −3.470133515730959, −2.543493367727356, −2.153833472786316, −1.725003938367976, −0.4754919863275742,
0.4754919863275742, 1.725003938367976, 2.153833472786316, 2.543493367727356, 3.470133515730959, 4.164015334703001, 4.576154798164534, 5.156125631289420, 5.995127233933316, 6.122202465482822, 6.730537469121942, 7.327999363030190, 7.975248492582366, 8.590935245798473, 9.044164543791231, 9.795735029500446, 10.21216470596385, 10.49715254205282, 11.38127058263251, 11.67714470422617, 12.17773162745189, 12.91689400644592, 13.29655252366993, 13.48349053374247, 14.18296903233057
