| L(s) = 1 | − 4·5-s − 7-s + 3·11-s + 5·13-s + 17-s + 6·19-s + 5·23-s + 11·25-s − 7·29-s + 8·31-s + 4·35-s − 2·37-s − 6·41-s + 11·43-s − 6·49-s + 10·53-s − 12·55-s − 20·65-s − 2·67-s + 2·71-s − 9·73-s − 3·77-s − 79-s − 11·83-s − 4·85-s + 8·89-s − 5·91-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 0.377·7-s + 0.904·11-s + 1.38·13-s + 0.242·17-s + 1.37·19-s + 1.04·23-s + 11/5·25-s − 1.29·29-s + 1.43·31-s + 0.676·35-s − 0.328·37-s − 0.937·41-s + 1.67·43-s − 6/7·49-s + 1.37·53-s − 1.61·55-s − 2.48·65-s − 0.244·67-s + 0.237·71-s − 1.05·73-s − 0.341·77-s − 0.112·79-s − 1.20·83-s − 0.433·85-s + 0.847·89-s − 0.524·91-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
| \(L(1)\) |
\(\approx\) |
\(1.701411358\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.701411358\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 + 7 T + p T^{2} \) | 1.29.h |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 83 | \( 1 + 11 T + p T^{2} \) | 1.83.l |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−16.29991574811083, −15.84823758694019, −15.46654484138561, −14.86854243119378, −14.27263602310469, −13.53685436294613, −13.04054527967844, −12.22719017445922, −11.83367617163699, −11.31560853110246, −10.93572938640058, −10.07643594238692, −9.268597831173592, −8.743755436210137, −8.224422021336422, −7.431168161534105, −7.088509578627604, −6.313441029490057, −5.547863610926289, −4.652675050725748, −3.940805136695631, −3.482385133205065, −2.928488682404427, −1.341154212669863, −0.6866683823373107,
0.6866683823373107, 1.341154212669863, 2.928488682404427, 3.482385133205065, 3.940805136695631, 4.652675050725748, 5.547863610926289, 6.313441029490057, 7.088509578627604, 7.431168161534105, 8.224422021336422, 8.743755436210137, 9.268597831173592, 10.07643594238692, 10.93572938640058, 11.31560853110246, 11.83367617163699, 12.22719017445922, 13.04054527967844, 13.53685436294613, 14.27263602310469, 14.86854243119378, 15.46654484138561, 15.84823758694019, 16.29991574811083
