| L(s) = 1 | + 3-s + 9-s + 4·11-s − 13-s + 2·17-s + 4·19-s − 4·23-s − 5·25-s + 27-s + 6·31-s + 4·33-s + 2·37-s − 39-s + 6·41-s + 12·47-s + 2·51-s − 8·53-s + 4·57-s − 12·59-s + 6·61-s − 4·69-s + 12·71-s − 6·73-s − 5·75-s + 14·79-s + 81-s + 4·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 1.20·11-s − 0.277·13-s + 0.485·17-s + 0.917·19-s − 0.834·23-s − 25-s + 0.192·27-s + 1.07·31-s + 0.696·33-s + 0.328·37-s − 0.160·39-s + 0.937·41-s + 1.75·47-s + 0.280·51-s − 1.09·53-s + 0.529·57-s − 1.56·59-s + 0.768·61-s − 0.481·69-s + 1.42·71-s − 0.702·73-s − 0.577·75-s + 1.57·79-s + 1/9·81-s + 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 244608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 244608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.662787236\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.662787236\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−12.87679774585177, −12.25415295373110, −12.00027315946127, −11.66794455691456, −11.03316795119809, −10.48862401088473, −9.938098166304895, −9.585033112169722, −9.173535424858283, −8.811231596321254, −8.009403200160044, −7.724960489863020, −7.432479743841329, −6.609471778241019, −6.247645572510825, −5.815054165217700, −5.099669879543239, −4.555876551973798, −4.004087787422386, −3.591781995367439, −3.066651027262387, −2.337154249014673, −1.893179089603673, −1.112210658924677, −0.6305903110616320,
0.6305903110616320, 1.112210658924677, 1.893179089603673, 2.337154249014673, 3.066651027262387, 3.591781995367439, 4.004087787422386, 4.555876551973798, 5.099669879543239, 5.815054165217700, 6.247645572510825, 6.609471778241019, 7.432479743841329, 7.724960489863020, 8.009403200160044, 8.811231596321254, 9.173535424858283, 9.585033112169722, 9.938098166304895, 10.48862401088473, 11.03316795119809, 11.66794455691456, 12.00027315946127, 12.25415295373110, 12.87679774585177
