| L(s) = 1 | − 3·3-s + 2·5-s + 7-s + 6·9-s + 5·13-s − 6·15-s + 8·17-s + 3·19-s − 3·21-s − 23-s − 25-s − 9·27-s − 8·29-s + 8·31-s + 2·35-s − 2·37-s − 15·39-s − 2·41-s + 2·43-s + 12·45-s + 4·47-s + 49-s − 24·51-s + 12·53-s − 9·57-s + 15·59-s + 10·61-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.894·5-s + 0.377·7-s + 2·9-s + 1.38·13-s − 1.54·15-s + 1.94·17-s + 0.688·19-s − 0.654·21-s − 0.208·23-s − 1/5·25-s − 1.73·27-s − 1.48·29-s + 1.43·31-s + 0.338·35-s − 0.328·37-s − 2.40·39-s − 0.312·41-s + 0.304·43-s + 1.78·45-s + 0.583·47-s + 1/7·49-s − 3.36·51-s + 1.64·53-s − 1.19·57-s + 1.95·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.015938821\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.015938821\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 - 8 T + p T^{2} \) | 1.17.ai |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 15 T + p T^{2} \) | 1.59.ap |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 3 T + p T^{2} \) | 1.79.ad |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
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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
−16.21534660639505, −16.00953962718958, −15.17061669915619, −14.50962916545173, −13.73223698281027, −13.43106791633218, −12.73197082739011, −11.96544852612342, −11.78318768308286, −11.07565239268396, −10.56998400361288, −9.897042014881676, −9.707809789186490, −8.648530846401388, −7.942398679778847, −7.219885186333235, −6.547813893202393, −5.882595439764761, −5.469993832818268, −5.236323347752149, −4.074532067857358, −3.544058945749321, −2.228641100796786, −1.249228197628446, −0.8457658698312859,
0.8457658698312859, 1.249228197628446, 2.228641100796786, 3.544058945749321, 4.074532067857358, 5.236323347752149, 5.469993832818268, 5.882595439764761, 6.547813893202393, 7.219885186333235, 7.942398679778847, 8.648530846401388, 9.707809789186490, 9.897042014881676, 10.56998400361288, 11.07565239268396, 11.78318768308286, 11.96544852612342, 12.73197082739011, 13.43106791633218, 13.73223698281027, 14.50962916545173, 15.17061669915619, 16.00953962718958, 16.21534660639505
