| L(s) = 1 | + 3-s + 5-s + 3·7-s − 2·9-s + 5·11-s + 4·13-s + 15-s − 4·17-s + 8·19-s + 3·21-s − 4·23-s + 25-s − 5·27-s + 4·29-s − 2·31-s + 5·33-s + 3·35-s + 37-s + 4·39-s − 5·41-s + 6·43-s − 2·45-s − 9·47-s + 2·49-s − 4·51-s + 3·53-s + 5·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.13·7-s − 2/3·9-s + 1.50·11-s + 1.10·13-s + 0.258·15-s − 0.970·17-s + 1.83·19-s + 0.654·21-s − 0.834·23-s + 1/5·25-s − 0.962·27-s + 0.742·29-s − 0.359·31-s + 0.870·33-s + 0.507·35-s + 0.164·37-s + 0.640·39-s − 0.780·41-s + 0.914·43-s − 0.298·45-s − 1.31·47-s + 2/7·49-s − 0.560·51-s + 0.412·53-s + 0.674·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.152386725\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.152386725\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 37 | \( 1 - T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 + 15 T + p T^{2} \) | 1.73.p |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + 11 T + p T^{2} \) | 1.83.l |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−8.759558258871026004521818336444, −8.186433202743037417632015261638, −7.34245287835085792063157381447, −6.39344195974738925783632861954, −5.76546523585426885170756070202, −4.82959370676374676230940604175, −3.92637670973397291184766507979, −3.13141667245378763659368593597, −1.95088846348121714736358118750, −1.19620939459244487628546154310,
1.19620939459244487628546154310, 1.95088846348121714736358118750, 3.13141667245378763659368593597, 3.92637670973397291184766507979, 4.82959370676374676230940604175, 5.76546523585426885170756070202, 6.39344195974738925783632861954, 7.34245287835085792063157381447, 8.186433202743037417632015261638, 8.759558258871026004521818336444
