| L(s) = 1 | + 2-s + 4-s + 5-s + 7-s + 8-s + 10-s − 2·11-s − 6·13-s + 14-s + 16-s + 6·17-s − 2·19-s + 20-s − 2·22-s + 7·23-s − 4·25-s − 6·26-s + 28-s − 4·31-s + 32-s + 6·34-s + 35-s − 2·38-s + 40-s + 2·41-s + 6·43-s − 2·44-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 − 0.603·11-s − 1.66·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.458·19-s + 0.223·20-s − 0.426·22-s + 1.45·23-s − 4/5·25-s − 1.17·26-s + 0.188·28-s − 0.718·31-s + 0.176·32-s + 1.02·34-s + 0.169·35-s − 0.324·38-s + 0.158·40-s + 0.312·41-s + 0.914·43-s − 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136242 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.240879025\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.240879025\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 29 | \( 1 \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 7 T + p T^{2} \) | 1.23.ah |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 5 T + p T^{2} \) | 1.53.af |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - 12 T + p T^{2} \) | 1.61.am |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−13.25849114264835, −12.95051967409261, −12.64970459106987, −12.02417574630493, −11.66720620269130, −11.09722887283332, −10.52239346064709, −10.17841015030831, −9.538240929009850, −9.327324132581465, −8.413321464976790, −7.942742451048277, −7.396447472407776, −7.153674534541231, −6.434537876044075, −5.744006099029200, −5.402929977280725, −4.907129084679559, −4.562440939702443, −3.649156278562473, −3.244249698113448, −2.400431923124619, −2.228414204648202, −1.342192087288448, −0.5360086706146842,
0.5360086706146842, 1.342192087288448, 2.228414204648202, 2.400431923124619, 3.244249698113448, 3.649156278562473, 4.562440939702443, 4.907129084679559, 5.402929977280725, 5.744006099029200, 6.434537876044075, 7.153674534541231, 7.396447472407776, 7.942742451048277, 8.413321464976790, 9.327324132581465, 9.538240929009850, 10.17841015030831, 10.52239346064709, 11.09722887283332, 11.66720620269130, 12.02417574630493, 12.64970459106987, 12.95051967409261, 13.25849114264835
