| L(s) = 1 | + 2·2-s − 3-s + 2·4-s − 2·6-s + 5·7-s − 2·9-s + 3·11-s − 2·12-s + 2·13-s + 10·14-s − 4·16-s + 4·17-s − 4·18-s − 4·19-s − 5·21-s + 6·22-s + 2·23-s + 4·26-s + 5·27-s + 10·28-s + 2·29-s − 8·32-s − 3·33-s + 8·34-s − 4·36-s + 37-s − 8·38-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s − 0.816·6-s + 1.88·7-s − 2/3·9-s + 0.904·11-s − 0.577·12-s + 0.554·13-s + 2.67·14-s − 16-s + 0.970·17-s − 0.942·18-s − 0.917·19-s − 1.09·21-s + 1.27·22-s + 0.417·23-s + 0.784·26-s + 0.962·27-s + 1.88·28-s + 0.371·29-s − 1.41·32-s − 0.522·33-s + 1.37·34-s − 2/3·36-s + 0.164·37-s − 1.29·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.213748806\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.213748806\) |
| \(L(\frac{3}{2})\) |
|
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 | 5 | \( 1 \) | |
| 37 | \( 1 - T \) | |
| good | 2 | \( 1 - p T + p T^{2} \) | 1.2.ac |
| 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 - 5 T + p T^{2} \) | 1.7.af |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 41 | \( 1 - 7 T + p T^{2} \) | 1.41.ah |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + 11 T + p T^{2} \) | 1.47.l |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 + 16 T + p T^{2} \) | 1.67.q |
| 71 | \( 1 + 15 T + p T^{2} \) | 1.71.p |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−10.60855403390974587720352424268, −9.037708122147747799705982873278, −8.385038889128347053395819374736, −7.32877070600283154035094752201, −6.12609005306951811640483977590, −5.66470469165381738162827215332, −4.70424603737437836314377918089, −4.15459429389693587359434143709, −2.84209119849355748547672875930, −1.40770936121954750346064260205,
1.40770936121954750346064260205, 2.84209119849355748547672875930, 4.15459429389693587359434143709, 4.70424603737437836314377918089, 5.66470469165381738162827215332, 6.12609005306951811640483977590, 7.32877070600283154035094752201, 8.385038889128347053395819374736, 9.037708122147747799705982873278, 10.60855403390974587720352424268
