| L(s) = 1 | + 3-s − 2·5-s − 4·7-s − 2·9-s − 7·13-s − 2·15-s + 4·17-s − 6·19-s − 4·21-s + 23-s − 25-s − 5·27-s − 5·29-s − 3·31-s + 8·35-s + 2·37-s − 7·39-s + 9·41-s + 8·43-s + 4·45-s + 47-s + 9·49-s + 4·51-s − 6·53-s − 6·57-s + 8·59-s + 10·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s − 1.51·7-s − 2/3·9-s − 1.94·13-s − 0.516·15-s + 0.970·17-s − 1.37·19-s − 0.872·21-s + 0.208·23-s − 1/5·25-s − 0.962·27-s − 0.928·29-s − 0.538·31-s + 1.35·35-s + 0.328·37-s − 1.12·39-s + 1.40·41-s + 1.21·43-s + 0.596·45-s + 0.145·47-s + 9/7·49-s + 0.560·51-s − 0.824·53-s − 0.794·57-s + 1.04·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 + 7 T + p T^{2} \) | 1.13.h |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - T + p T^{2} \) | 1.47.ab |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 13 T + p T^{2} \) | 1.71.an |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−14.77303010873679, −14.57964980303043, −14.05367636929852, −13.21566067125431, −12.77718943097416, −12.39738422428329, −12.03388450867082, −11.21564762363361, −10.86111540046859, −9.970377590470358, −9.643962959906735, −9.242919860413263, −8.605951592982335, −7.843818560998189, −7.615562485513774, −7.004165644802579, −6.378737024956694, −5.696074329475715, −5.194853058251125, −4.200554674084508, −3.835142738851802, −3.199535163940220, −2.540196707508551, −2.180699208346766, −0.6143075373678353, 0,
0.6143075373678353, 2.180699208346766, 2.540196707508551, 3.199535163940220, 3.835142738851802, 4.200554674084508, 5.194853058251125, 5.696074329475715, 6.378737024956694, 7.004165644802579, 7.615562485513774, 7.843818560998189, 8.605951592982335, 9.242919860413263, 9.643962959906735, 9.970377590470358, 10.86111540046859, 11.21564762363361, 12.03388450867082, 12.39738422428329, 12.77718943097416, 13.21566067125431, 14.05367636929852, 14.57964980303043, 14.77303010873679
