| L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s − 11-s + 13-s + 16-s − 4·17-s + 2·19-s − 20-s + 22-s + 23-s − 4·25-s − 26-s + 9·29-s + 4·31-s − 32-s + 4·34-s − 6·37-s − 2·38-s + 40-s + 41-s + 11·43-s − 44-s − 46-s + 4·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s − 0.301·11-s + 0.277·13-s + 1/4·16-s − 0.970·17-s + 0.458·19-s − 0.223·20-s + 0.213·22-s + 0.208·23-s − 4/5·25-s − 0.196·26-s + 1.67·29-s + 0.718·31-s − 0.176·32-s + 0.685·34-s − 0.986·37-s − 0.324·38-s + 0.158·40-s + 0.156·41-s + 1.67·43-s − 0.150·44-s − 0.147·46-s + 0.565·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.236535340\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.236535340\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - T + p T^{2} \) | 1.41.ab |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 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
−13.48374990233994, −13.10060215967006, −12.26651590939125, −12.09379730045119, −11.55601698616496, −11.03597874415014, −10.54367305946928, −10.22062152329950, −9.584204138140472, −9.058937146796846, −8.599599212664902, −8.200707024461047, −7.655183969861011, −7.117790511227452, −6.740046356657428, −6.026840203785413, −5.663982746984110, −4.822471234309752, −4.359723857553712, −3.785096605044454, −2.979420250805767, −2.597879374737181, −1.849455734622633, −1.080421153146630, −0.4252493609992378,
0.4252493609992378, 1.080421153146630, 1.849455734622633, 2.597879374737181, 2.979420250805767, 3.785096605044454, 4.359723857553712, 4.822471234309752, 5.663982746984110, 6.026840203785413, 6.740046356657428, 7.117790511227452, 7.655183969861011, 8.200707024461047, 8.599599212664902, 9.058937146796846, 9.584204138140472, 10.22062152329950, 10.54367305946928, 11.03597874415014, 11.55601698616496, 12.09379730045119, 12.26651590939125, 13.10060215967006, 13.48374990233994
