| L(s) = 1 | + 3-s − 7-s − 2·9-s − 3·11-s − 13-s + 4·17-s − 2·19-s − 21-s − 23-s − 5·25-s − 5·27-s + 4·29-s − 9·31-s − 3·33-s + 3·37-s − 39-s − 5·41-s − 4·43-s − 9·47-s + 49-s + 4·51-s − 4·53-s − 2·57-s − 10·59-s + 5·61-s + 2·63-s − 11·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s − 2/3·9-s − 0.904·11-s − 0.277·13-s + 0.970·17-s − 0.458·19-s − 0.218·21-s − 0.208·23-s − 25-s − 0.962·27-s + 0.742·29-s − 1.61·31-s − 0.522·33-s + 0.493·37-s − 0.160·39-s − 0.780·41-s − 0.609·43-s − 1.31·47-s + 1/7·49-s + 0.560·51-s − 0.549·53-s − 0.264·57-s − 1.30·59-s + 0.640·61-s + 0.251·63-s − 1.34·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 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 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + 11 T + p T^{2} \) | 1.67.l |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 - 5 T + p T^{2} \) | 1.79.af |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−9.180246455905551274823414752906, −8.090946695882983429394031186652, −7.86106901905057460373915465638, −6.68837221260609580260058636515, −5.75343602406759559067566546527, −5.02101645933316463421059800388, −3.70736881607893429019058281380, −2.97429105377228843634074901438, −1.95742675999762562132181188496, 0,
1.95742675999762562132181188496, 2.97429105377228843634074901438, 3.70736881607893429019058281380, 5.02101645933316463421059800388, 5.75343602406759559067566546527, 6.68837221260609580260058636515, 7.86106901905057460373915465638, 8.090946695882983429394031186652, 9.180246455905551274823414752906
