| L(s) = 1 | − 2·5-s + 2·7-s − 2·11-s + 6·13-s − 4·19-s − 25-s − 2·29-s − 4·31-s − 4·35-s − 8·37-s + 6·41-s − 3·49-s − 6·53-s + 4·55-s + 12·59-s − 12·61-s − 12·65-s + 8·67-s − 10·73-s − 4·77-s + 14·79-s − 6·83-s + 12·91-s + 8·95-s − 10·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.755·7-s − 0.603·11-s + 1.66·13-s − 0.917·19-s − 1/5·25-s − 0.371·29-s − 0.718·31-s − 0.676·35-s − 1.31·37-s + 0.937·41-s − 3/7·49-s − 0.824·53-s + 0.539·55-s + 1.56·59-s − 1.53·61-s − 1.48·65-s + 0.977·67-s − 1.17·73-s − 0.455·77-s + 1.57·79-s − 0.658·83-s + 1.25·91-s + 0.820·95-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 23 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−13.48251469637089, −13.03342580930777, −12.70304228140806, −12.07322370002776, −11.54118015253611, −11.18794523532763, −10.78285066039167, −10.47382311854051, −9.746686903113578, −9.008106022821381, −8.704110410866627, −8.157702882180252, −7.871956488420931, −7.347793139629637, −6.725135692038774, −6.159920729622793, −5.644831239712022, −5.095123895461918, −4.468026310393998, −3.970443304301848, −3.551923541953430, −2.957499579247281, −2.017811842667530, −1.649039468956990, −0.7692707748145386, 0,
0.7692707748145386, 1.649039468956990, 2.017811842667530, 2.957499579247281, 3.551923541953430, 3.970443304301848, 4.468026310393998, 5.095123895461918, 5.644831239712022, 6.159920729622793, 6.725135692038774, 7.347793139629637, 7.871956488420931, 8.157702882180252, 8.704110410866627, 9.008106022821381, 9.746686903113578, 10.47382311854051, 10.78285066039167, 11.18794523532763, 11.54118015253611, 12.07322370002776, 12.70304228140806, 13.03342580930777, 13.48251469637089
