| L(s) = 1 | + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s − 11-s + 13-s − 14-s + 16-s + 2·17-s − 20-s − 22-s + 25-s + 26-s − 28-s + 32-s + 2·34-s + 35-s + 10·37-s − 40-s − 2·41-s + 6·43-s − 44-s + 49-s + 50-s + 52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s + 0.353·8-s − 0.316·10-s − 0.301·11-s + 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.223·20-s − 0.213·22-s + 1/5·25-s + 0.196·26-s − 0.188·28-s + 0.176·32-s + 0.342·34-s + 0.169·35-s + 1.64·37-s − 0.158·40-s − 0.312·41-s + 0.914·43-s − 0.150·44-s + 1/7·49-s + 0.141·50-s + 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90090 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90090 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.707988455\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.707988455\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| good | 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 - 12 T + p T^{2} \) | 1.73.am |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−13.88369443021228, −13.25573771800379, −12.95402708013352, −12.46555857315115, −11.93891828807214, −11.48685541567013, −11.02381399236232, −10.47768657199314, −9.982762777546969, −9.426918009586299, −8.857664425547611, −8.148211792877500, −7.810540428193498, −7.233082326269420, −6.624392896308263, −6.223391062173649, −5.460996457022413, −5.217570048687765, −4.341304829440214, −3.935022430696729, −3.425199973314401, −2.671193058776161, −2.287254304324896, −1.229470214759825, −0.5836959336524173,
0.5836959336524173, 1.229470214759825, 2.287254304324896, 2.671193058776161, 3.425199973314401, 3.935022430696729, 4.341304829440214, 5.217570048687765, 5.460996457022413, 6.223391062173649, 6.624392896308263, 7.233082326269420, 7.810540428193498, 8.148211792877500, 8.857664425547611, 9.426918009586299, 9.982762777546969, 10.47768657199314, 11.02381399236232, 11.48685541567013, 11.93891828807214, 12.46555857315115, 12.95402708013352, 13.25573771800379, 13.88369443021228
