| L(s) = 1 | − 3-s + 7-s − 2·9-s − 11-s − 4·13-s + 2·17-s − 8·19-s − 21-s + 8·23-s + 5·27-s + 4·29-s − 9·31-s + 33-s + 3·37-s + 4·39-s + 12·43-s − 12·47-s − 6·49-s − 2·51-s − 7·53-s + 8·57-s + 8·61-s − 2·63-s + 10·67-s − 8·69-s − 12·71-s − 2·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s − 2/3·9-s − 0.301·11-s − 1.10·13-s + 0.485·17-s − 1.83·19-s − 0.218·21-s + 1.66·23-s + 0.962·27-s + 0.742·29-s − 1.61·31-s + 0.174·33-s + 0.493·37-s + 0.640·39-s + 1.82·43-s − 1.75·47-s − 6/7·49-s − 0.280·51-s − 0.961·53-s + 1.05·57-s + 1.02·61-s − 0.251·63-s + 1.22·67-s − 0.963·69-s − 1.42·71-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6707498128\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6707498128\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 79 | \( 1 + T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + 9 T + p T^{2} \) | 1.31.j |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 7 T + p T^{2} \) | 1.53.h |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−13.23935133520266, −12.98592389922768, −12.58514595383923, −12.07305114588020, −11.53634689059603, −11.02648071054119, −10.73318545776361, −10.35143719641979, −9.454539578785775, −9.278669383566873, −8.561953117835009, −8.082618703378523, −7.674797841084300, −6.914277526506078, −6.603270670426302, −5.991169047991265, −5.321823694245378, −5.045036248137084, −4.526558348726548, −3.865817293084790, −3.022913083082335, −2.614146400478696, −1.959408218875127, −1.139366305220705, −0.2720844771301744,
0.2720844771301744, 1.139366305220705, 1.959408218875127, 2.614146400478696, 3.022913083082335, 3.865817293084790, 4.526558348726548, 5.045036248137084, 5.321823694245378, 5.991169047991265, 6.603270670426302, 6.914277526506078, 7.674797841084300, 8.082618703378523, 8.561953117835009, 9.278669383566873, 9.454539578785775, 10.35143719641979, 10.73318545776361, 11.02648071054119, 11.53634689059603, 12.07305114588020, 12.58514595383923, 12.98592389922768, 13.23935133520266
