| L(s) = 1 | + 3-s + 7-s + 9-s − 11-s − 13-s + 4·17-s + 3·19-s + 21-s − 2·23-s + 27-s + 5·29-s − 33-s − 9·37-s − 39-s − 10·43-s − 5·47-s + 49-s + 4·51-s + 6·53-s + 3·57-s + 13·59-s + 6·61-s + 63-s + 67-s − 2·69-s + 9·73-s − 77-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.277·13-s + 0.970·17-s + 0.688·19-s + 0.218·21-s − 0.417·23-s + 0.192·27-s + 0.928·29-s − 0.174·33-s − 1.47·37-s − 0.160·39-s − 1.52·43-s − 0.729·47-s + 1/7·49-s + 0.560·51-s + 0.824·53-s + 0.397·57-s + 1.69·59-s + 0.768·61-s + 0.125·63-s + 0.122·67-s − 0.240·69-s + 1.05·73-s − 0.113·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.040953731\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.040953731\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| good | 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 9 T + p T^{2} \) | 1.37.j |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + 5 T + p T^{2} \) | 1.47.f |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 13 T + p T^{2} \) | 1.59.an |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - T + p T^{2} \) | 1.67.ab |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 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
−15.40295279479486, −14.83812276536857, −14.45602809441591, −13.77737900175040, −13.56009903047305, −12.76412162102230, −12.12958904467853, −11.85363188443309, −11.09970114326566, −10.37988967259604, −9.939448744305407, −9.533064523607032, −8.618434725676697, −8.255434353350412, −7.783843304653228, −6.990607232701452, −6.650696393535898, −5.492711113089003, −5.269986251758172, −4.469496540852121, −3.631992104917379, −3.165928559481153, −2.321750911683575, −1.613600475477110, −0.6903129954087803,
0.6903129954087803, 1.613600475477110, 2.321750911683575, 3.165928559481153, 3.631992104917379, 4.469496540852121, 5.269986251758172, 5.492711113089003, 6.650696393535898, 6.990607232701452, 7.783843304653228, 8.255434353350412, 8.618434725676697, 9.533064523607032, 9.939448744305407, 10.37988967259604, 11.09970114326566, 11.85363188443309, 12.12958904467853, 12.76412162102230, 13.56009903047305, 13.77737900175040, 14.45602809441591, 14.83812276536857, 15.40295279479486
