| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s + 4·11-s − 13-s + 15-s + 7·19-s + 21-s − 6·23-s + 25-s − 27-s + 2·29-s + 3·31-s − 4·33-s + 35-s + 11·37-s + 39-s − 43-s − 45-s − 2·47-s − 6·49-s − 6·53-s − 4·55-s − 7·57-s + 6·59-s + 13·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.277·13-s + 0.258·15-s + 1.60·19-s + 0.218·21-s − 1.25·23-s + 1/5·25-s − 0.192·27-s + 0.371·29-s + 0.538·31-s − 0.696·33-s + 0.169·35-s + 1.80·37-s + 0.160·39-s − 0.152·43-s − 0.149·45-s − 0.291·47-s − 6/7·49-s − 0.824·53-s − 0.539·55-s − 0.927·57-s + 0.781·59-s + 1.66·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 - 11 T + p T^{2} \) | 1.37.al |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 13 T + p T^{2} \) | 1.61.an |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
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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
−14.47264605765138, −13.99762281537588, −13.27442914772283, −12.90893445693297, −12.23393193600480, −11.72838592019331, −11.60723229412612, −11.11032085196476, −10.15450235613166, −9.906999138320578, −9.489988527967636, −8.827219908558474, −8.188163089631609, −7.658531148703364, −7.121983869321665, −6.604979123707035, −6.019642259753107, −5.655046046467487, −4.756849790243575, −4.390975863724186, −3.721209432085145, −3.179023639915702, −2.439858907017735, −1.448330144039368, −0.9246155768924138, 0,
0.9246155768924138, 1.448330144039368, 2.439858907017735, 3.179023639915702, 3.721209432085145, 4.390975863724186, 4.756849790243575, 5.655046046467487, 6.019642259753107, 6.604979123707035, 7.121983869321665, 7.658531148703364, 8.188163089631609, 8.827219908558474, 9.489988527967636, 9.906999138320578, 10.15450235613166, 11.11032085196476, 11.60723229412612, 11.72838592019331, 12.23393193600480, 12.90893445693297, 13.27442914772283, 13.99762281537588, 14.47264605765138
