| L(s) = 1 | + 3-s + 5-s + 9-s + 2·11-s + 13-s + 15-s + 8·19-s − 3·23-s − 4·25-s + 27-s + 29-s + 8·31-s + 2·33-s + 9·37-s + 39-s − 41-s + 4·43-s + 45-s + 7·47-s + 53-s + 2·55-s + 8·57-s + 2·59-s + 8·61-s + 65-s − 5·67-s − 3·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.603·11-s + 0.277·13-s + 0.258·15-s + 1.83·19-s − 0.625·23-s − 4/5·25-s + 0.192·27-s + 0.185·29-s + 1.43·31-s + 0.348·33-s + 1.47·37-s + 0.160·39-s − 0.156·41-s + 0.609·43-s + 0.149·45-s + 1.02·47-s + 0.137·53-s + 0.269·55-s + 1.05·57-s + 0.260·59-s + 1.02·61-s + 0.124·65-s − 0.610·67-s − 0.361·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 385728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.065771307\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.065771307\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 41 | \( 1 + T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 9 T + p T^{2} \) | 1.37.aj |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 13 T + p T^{2} \) | 1.97.an |
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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
−12.23888025416623, −12.12861049419530, −11.63725081814620, −11.14394171929723, −10.63911730536175, −9.942280833697636, −9.753378829261177, −9.410583314675099, −8.915414055770807, −8.286814599261898, −7.972163600453575, −7.491903688907762, −7.004022021546295, −6.412340664659668, −6.031303004201117, −5.500010815486491, −5.051784884800354, −4.235050323196969, −4.048988919941101, −3.379387397775768, −2.818322393550035, −2.385273321975932, −1.735405031758056, −1.063946879702164, −0.6895361835882663,
0.6895361835882663, 1.063946879702164, 1.735405031758056, 2.385273321975932, 2.818322393550035, 3.379387397775768, 4.048988919941101, 4.235050323196969, 5.051784884800354, 5.500010815486491, 6.031303004201117, 6.412340664659668, 7.004022021546295, 7.491903688907762, 7.972163600453575, 8.286814599261898, 8.915414055770807, 9.410583314675099, 9.753378829261177, 9.942280833697636, 10.63911730536175, 11.14394171929723, 11.63725081814620, 12.12861049419530, 12.23888025416623
