| L(s) = 1 | − 5-s + 7-s + 5·11-s + 13-s − 5·17-s + 5·19-s − 7·23-s − 4·25-s − 29-s − 6·31-s − 35-s + 11·37-s + 2·41-s + 11·43-s − 8·47-s + 49-s + 2·53-s − 5·55-s − 10·59-s + 5·61-s − 65-s − 12·67-s + 14·71-s − 7·73-s + 5·77-s − 6·79-s − 2·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s + 1.50·11-s + 0.277·13-s − 1.21·17-s + 1.14·19-s − 1.45·23-s − 4/5·25-s − 0.185·29-s − 1.07·31-s − 0.169·35-s + 1.80·37-s + 0.312·41-s + 1.67·43-s − 1.16·47-s + 1/7·49-s + 0.274·53-s − 0.674·55-s − 1.30·59-s + 0.640·61-s − 0.124·65-s − 1.46·67-s + 1.66·71-s − 0.819·73-s + 0.569·77-s − 0.675·79-s − 0.219·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 7 T + p T^{2} \) | 1.23.h |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 - 11 T + p T^{2} \) | 1.37.al |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−15.68509310450589, −14.99767417608262, −14.44592734342937, −14.06550597403488, −13.55475689657043, −12.84923797892712, −12.29797872986451, −11.58533857245645, −11.44108100126457, −10.94079133046923, −10.04306875477148, −9.449885530076586, −9.102256767005950, −8.442121432440291, −7.677444831479939, −7.451005419982705, −6.530193791637310, −6.089521430743875, −5.493564337900786, −4.472948649527129, −4.116372167847159, −3.619597437483690, −2.633241424545070, −1.811803503709229, −1.125079632264748, 0,
1.125079632264748, 1.811803503709229, 2.633241424545070, 3.619597437483690, 4.116372167847159, 4.472948649527129, 5.493564337900786, 6.089521430743875, 6.530193791637310, 7.451005419982705, 7.677444831479939, 8.442121432440291, 9.102256767005950, 9.449885530076586, 10.04306875477148, 10.94079133046923, 11.44108100126457, 11.58533857245645, 12.29797872986451, 12.84923797892712, 13.55475689657043, 14.06550597403488, 14.44592734342937, 14.99767417608262, 15.68509310450589
