| L(s) = 1 | − 3-s + 7-s + 9-s − 11-s + 2·13-s − 3·17-s + 19-s − 21-s − 3·23-s − 27-s − 29-s − 2·31-s + 33-s + 11·37-s − 2·39-s + 3·41-s + 4·43-s − 3·47-s − 6·49-s + 3·51-s − 57-s + 9·59-s − 10·61-s + 63-s + 4·67-s + 3·69-s − 3·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s − 0.727·17-s + 0.229·19-s − 0.218·21-s − 0.625·23-s − 0.192·27-s − 0.185·29-s − 0.359·31-s + 0.174·33-s + 1.80·37-s − 0.320·39-s + 0.468·41-s + 0.609·43-s − 0.437·47-s − 6/7·49-s + 0.420·51-s − 0.132·57-s + 1.17·59-s − 1.28·61-s + 0.125·63-s + 0.488·67-s + 0.361·69-s − 0.356·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 382800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 382800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| 29 | \( 1 + T \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 11 T + p T^{2} \) | 1.37.al |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 7 T + p T^{2} \) | 1.79.ah |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 19 T + p T^{2} \) | 1.97.t |
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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
−12.67682104355231, −12.26390178867056, −11.65762388948963, −11.24348902370097, −11.07592050545880, −10.56504588260064, −9.942599864407118, −9.670682723917026, −9.032602641975369, −8.666845766831143, −8.028997160856477, −7.672346911086315, −7.286905385422039, −6.503907017936520, −6.274883972894283, −5.755479712863835, −5.299794308475529, −4.573865466914236, −4.466939996442221, −3.703366422718779, −3.263971158990372, −2.396890744541565, −2.091260199976363, −1.300042150232122, −0.7538413249465060, 0,
0.7538413249465060, 1.300042150232122, 2.091260199976363, 2.396890744541565, 3.263971158990372, 3.703366422718779, 4.466939996442221, 4.573865466914236, 5.299794308475529, 5.755479712863835, 6.274883972894283, 6.503907017936520, 7.286905385422039, 7.672346911086315, 8.028997160856477, 8.666845766831143, 9.032602641975369, 9.670682723917026, 9.942599864407118, 10.56504588260064, 11.07592050545880, 11.24348902370097, 11.65762388948963, 12.26390178867056, 12.67682104355231
