| L(s) = 1 | + 7-s + 11-s − 3·13-s − 3·17-s + 2·19-s − 23-s − 5·25-s − 3·29-s + 4·31-s − 2·37-s + 10·41-s + 5·43-s + 4·47-s − 6·49-s + 10·53-s − 4·61-s − 2·67-s − 6·71-s − 73-s + 77-s + 79-s − 9·83-s − 8·89-s − 3·91-s − 97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 0.301·11-s − 0.832·13-s − 0.727·17-s + 0.458·19-s − 0.208·23-s − 25-s − 0.557·29-s + 0.718·31-s − 0.328·37-s + 1.56·41-s + 0.762·43-s + 0.583·47-s − 6/7·49-s + 1.37·53-s − 0.512·61-s − 0.244·67-s − 0.712·71-s − 0.117·73-s + 0.113·77-s + 0.112·79-s − 0.987·83-s − 0.847·89-s − 0.314·91-s − 0.101·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 79 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + T + p T^{2} \) | 1.73.b |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−16.80649213184779, −16.11095005080694, −15.58522402487748, −15.05944901431157, −14.41185938350818, −13.97169116200688, −13.36708148946400, −12.70708010197468, −12.08301802149230, −11.58540314030454, −11.04460496541172, −10.32490423592195, −9.713682791145755, −9.161710661634346, −8.547571223463322, −7.686873462869366, −7.395948863917479, −6.546489465874500, −5.845739557421206, −5.220525144138526, −4.372551381462797, −3.939668461057333, −2.815966138848510, −2.190080862884812, −1.221347519730295, 0,
1.221347519730295, 2.190080862884812, 2.815966138848510, 3.939668461057333, 4.372551381462797, 5.220525144138526, 5.845739557421206, 6.546489465874500, 7.395948863917479, 7.686873462869366, 8.547571223463322, 9.161710661634346, 9.713682791145755, 10.32490423592195, 11.04460496541172, 11.58540314030454, 12.08301802149230, 12.70708010197468, 13.36708148946400, 13.97169116200688, 14.41185938350818, 15.05944901431157, 15.58522402487748, 16.11095005080694, 16.80649213184779
