| L(s) = 1 | + 3-s + 2·5-s − 7-s + 9-s + 2·13-s + 2·15-s + 4·17-s − 19-s − 21-s − 6·23-s − 25-s + 27-s + 6·29-s − 5·31-s − 2·35-s + 7·37-s + 2·39-s + 4·43-s + 2·45-s + 4·47-s − 6·49-s + 4·51-s + 6·53-s − 57-s − 6·59-s + 61-s − 63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s + 0.554·13-s + 0.516·15-s + 0.970·17-s − 0.229·19-s − 0.218·21-s − 1.25·23-s − 1/5·25-s + 0.192·27-s + 1.11·29-s − 0.898·31-s − 0.338·35-s + 1.15·37-s + 0.320·39-s + 0.609·43-s + 0.298·45-s + 0.583·47-s − 6/7·49-s + 0.560·51-s + 0.824·53-s − 0.132·57-s − 0.781·59-s + 0.128·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.315994330\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.315994330\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 - 9 T + p T^{2} \) | 1.79.aj |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 9 T + p T^{2} \) | 1.97.aj |
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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
−16.43407943376882, −15.84313060532942, −15.32356791062093, −14.53588527425852, −14.04760202668395, −13.78928221272395, −12.96933696574871, −12.64359672369939, −11.89851948088779, −11.22360754656497, −10.26144443482624, −10.15105110660959, −9.364507133139025, −8.963533238704423, −8.063740894141716, −7.729907312603713, −6.786845288778303, −6.093789961395689, −5.751962700830112, −4.813733228734925, −3.950700411931768, −3.349818262778784, −2.463530640446391, −1.828191560864866, −0.8233380484355209,
0.8233380484355209, 1.828191560864866, 2.463530640446391, 3.349818262778784, 3.950700411931768, 4.813733228734925, 5.751962700830112, 6.093789961395689, 6.786845288778303, 7.729907312603713, 8.063740894141716, 8.963533238704423, 9.364507133139025, 10.15105110660959, 10.26144443482624, 11.22360754656497, 11.89851948088779, 12.64359672369939, 12.96933696574871, 13.78928221272395, 14.04760202668395, 14.53588527425852, 15.32356791062093, 15.84313060532942, 16.43407943376882
