| L(s) = 1 | + 5-s + 2·7-s − 4·11-s − 13-s + 2·17-s + 2·19-s − 4·25-s + 5·29-s − 6·31-s + 2·35-s + 10·37-s + 9·41-s − 10·43-s + 12·47-s − 3·49-s + 5·53-s − 4·55-s + 6·59-s − 61-s − 65-s − 8·67-s + 6·71-s − 9·73-s − 8·77-s + 12·79-s − 10·83-s + 2·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.755·7-s − 1.20·11-s − 0.277·13-s + 0.485·17-s + 0.458·19-s − 4/5·25-s + 0.928·29-s − 1.07·31-s + 0.338·35-s + 1.64·37-s + 1.40·41-s − 1.52·43-s + 1.75·47-s − 3/7·49-s + 0.686·53-s − 0.539·55-s + 0.781·59-s − 0.128·61-s − 0.124·65-s − 0.977·67-s + 0.712·71-s − 1.05·73-s − 0.911·77-s + 1.35·79-s − 1.09·83-s + 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 5 T + p T^{2} \) | 1.53.af |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 10 T + p T^{2} \) | 1.83.k |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 - 15 T + p T^{2} \) | 1.97.ap |
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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
−12.95906312504888, −12.42730669242248, −11.98204782343039, −11.56035811421323, −10.93785030515702, −10.73177194877033, −10.12609052548628, −9.677081403441258, −9.414035161964470, −8.644179820124589, −8.235751136030101, −7.802217923465821, −7.408059461115745, −6.980648302101064, −6.181243817963887, −5.718699786806905, −5.405225249114433, −4.912718537241781, −4.323138849178864, −3.874976660911648, −3.030930882731259, −2.607802001475915, −2.140144877766056, −1.439387263543079, −0.8445715538047130, 0,
0.8445715538047130, 1.439387263543079, 2.140144877766056, 2.607802001475915, 3.030930882731259, 3.874976660911648, 4.323138849178864, 4.912718537241781, 5.405225249114433, 5.718699786806905, 6.181243817963887, 6.980648302101064, 7.408059461115745, 7.802217923465821, 8.235751136030101, 8.644179820124589, 9.414035161964470, 9.677081403441258, 10.12609052548628, 10.73177194877033, 10.93785030515702, 11.56035811421323, 11.98204782343039, 12.42730669242248, 12.95906312504888
