| L(s) = 1 | + 2·5-s − 3·7-s − 4·11-s − 4·13-s + 4·23-s − 25-s + 9·29-s + 4·31-s − 6·35-s − 7·37-s + 8·41-s − 4·43-s + 12·47-s + 2·49-s + 6·53-s − 8·55-s − 5·59-s − 2·61-s − 8·65-s + 2·67-s − 9·71-s − 13·73-s + 12·77-s − 12·79-s + 2·83-s + 14·89-s + 12·91-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.13·7-s − 1.20·11-s − 1.10·13-s + 0.834·23-s − 1/5·25-s + 1.67·29-s + 0.718·31-s − 1.01·35-s − 1.15·37-s + 1.24·41-s − 0.609·43-s + 1.75·47-s + 2/7·49-s + 0.824·53-s − 1.07·55-s − 0.650·59-s − 0.256·61-s − 0.992·65-s + 0.244·67-s − 1.06·71-s − 1.52·73-s + 1.36·77-s − 1.35·79-s + 0.219·83-s + 1.48·89-s + 1.25·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.482175898\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.482175898\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 397 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 5 T + p T^{2} \) | 1.59.f |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 + 13 T + p T^{2} \) | 1.73.n |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 17 T + p T^{2} \) | 1.97.r |
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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
−12.98173612250711, −12.54266671230703, −12.08489373153054, −11.73189198500094, −10.85352698863779, −10.42369723148062, −10.11011106237277, −9.866524065661174, −9.176210505908980, −8.880063959919667, −8.267299520386252, −7.617816144882435, −7.175227071985928, −6.794867480198331, −6.117731892034895, −5.788077541393099, −5.277418034948991, −4.688902283692953, −4.307379979041196, −3.331280351437385, −2.872861975179086, −2.559822676622158, −1.980282569553999, −1.069819523977338, −0.3569845482099398,
0.3569845482099398, 1.069819523977338, 1.980282569553999, 2.559822676622158, 2.872861975179086, 3.331280351437385, 4.307379979041196, 4.688902283692953, 5.277418034948991, 5.788077541393099, 6.117731892034895, 6.794867480198331, 7.175227071985928, 7.617816144882435, 8.267299520386252, 8.880063959919667, 9.176210505908980, 9.866524065661174, 10.11011106237277, 10.42369723148062, 10.85352698863779, 11.73189198500094, 12.08489373153054, 12.54266671230703, 12.98173612250711
