| L(s) = 1 | − 2-s + 4-s − 5-s + 7-s − 8-s + 10-s − 11-s − 13-s − 14-s + 16-s − 2·19-s − 20-s + 22-s + 8·23-s + 25-s + 26-s + 28-s + 2·29-s − 32-s − 35-s + 4·37-s + 2·38-s + 40-s − 2·41-s + 10·43-s − 44-s − 8·46-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 − 0.458·19-s − 0.223·20-s + 0.213·22-s + 1.66·23-s + 1/5·25-s + 0.196·26-s + 0.188·28-s + 0.371·29-s − 0.176·32-s − 0.169·35-s + 0.657·37-s + 0.324·38-s + 0.158·40-s − 0.312·41-s + 1.52·43-s − 0.150·44-s − 1.17·46-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
| \(L(1)\) |
\(\approx\) |
\(1.783599609\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.783599609\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 12 T + p T^{2} \) | 1.61.am |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−13.87437277993282, −13.31435317982161, −12.77596485588023, −12.35968423596899, −11.80458990137634, −11.26201528879566, −10.90172956774136, −10.45786252920229, −9.899031931473541, −9.293997923333891, −8.767458283158740, −8.482931697162476, −7.750739149651475, −7.407644330829805, −6.905742656286352, −6.358298485154407, −5.565046837509140, −5.194586826554563, −4.387359179143079, −4.006637346044001, −3.057306323963513, −2.639511902654950, −1.971781552722622, −1.040367825144603, −0.5687328558939420,
0.5687328558939420, 1.040367825144603, 1.971781552722622, 2.639511902654950, 3.057306323963513, 4.006637346044001, 4.387359179143079, 5.194586826554563, 5.565046837509140, 6.358298485154407, 6.905742656286352, 7.407644330829805, 7.750739149651475, 8.482931697162476, 8.767458283158740, 9.293997923333891, 9.899031931473541, 10.45786252920229, 10.90172956774136, 11.26201528879566, 11.80458990137634, 12.35968423596899, 12.77596485588023, 13.31435317982161, 13.87437277993282
