| L(s) = 1 | + 3-s + 7-s + 9-s − 3·11-s + 13-s − 17-s + 4·19-s + 21-s − 6·23-s + 27-s + 9·29-s + 5·31-s − 3·33-s + 8·37-s + 39-s + 2·41-s − 4·43-s + 3·47-s − 6·49-s − 51-s − 53-s + 4·57-s − 9·59-s + 7·61-s + 63-s − 15·67-s − 6·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.904·11-s + 0.277·13-s − 0.242·17-s + 0.917·19-s + 0.218·21-s − 1.25·23-s + 0.192·27-s + 1.67·29-s + 0.898·31-s − 0.522·33-s + 1.31·37-s + 0.160·39-s + 0.312·41-s − 0.609·43-s + 0.437·47-s − 6/7·49-s − 0.140·51-s − 0.137·53-s + 0.529·57-s − 1.17·59-s + 0.896·61-s + 0.125·63-s − 1.83·67-s − 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 + 15 T + p T^{2} \) | 1.67.p |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−13.67183917740415, −13.49035535162183, −12.88964803724957, −12.26325269931713, −11.92411779327165, −11.37585909824698, −10.82151039955258, −10.29248692546491, −9.872163127089123, −9.496858612765903, −8.731855389093344, −8.296714839694040, −7.960250431662148, −7.504604816248722, −6.899775591145863, −6.194960744317596, −5.862380156500721, −5.081632880994956, −4.566800846112687, −4.198906459381684, −3.345752981321537, −2.815893847891681, −2.420421778518249, −1.566951655954179, −0.9919972224502992, 0,
0.9919972224502992, 1.566951655954179, 2.420421778518249, 2.815893847891681, 3.345752981321537, 4.198906459381684, 4.566800846112687, 5.081632880994956, 5.862380156500721, 6.194960744317596, 6.899775591145863, 7.504604816248722, 7.960250431662148, 8.296714839694040, 8.731855389093344, 9.496858612765903, 9.872163127089123, 10.29248692546491, 10.82151039955258, 11.37585909824698, 11.92411779327165, 12.26325269931713, 12.88964803724957, 13.49035535162183, 13.67183917740415
