| L(s) = 1 | + 2-s + 4-s − 5-s + 8-s − 10-s + 3·11-s − 13-s + 16-s + 2·17-s − 20-s + 3·22-s − 6·23-s − 4·25-s − 26-s − 9·29-s + 5·31-s + 32-s + 2·34-s − 8·37-s − 40-s + 4·41-s + 3·44-s − 6·46-s − 4·50-s − 52-s − 53-s − 3·55-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s + 0.904·11-s − 0.277·13-s + 1/4·16-s + 0.485·17-s − 0.223·20-s + 0.639·22-s − 1.25·23-s − 4/5·25-s − 0.196·26-s − 1.67·29-s + 0.898·31-s + 0.176·32-s + 0.342·34-s − 1.31·37-s − 0.158·40-s + 0.624·41-s + 0.452·44-s − 0.884·46-s − 0.565·50-s − 0.138·52-s − 0.137·53-s − 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11466 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11466 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 - 7 T + p T^{2} \) | 1.59.ah |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 13 T + p T^{2} \) | 1.79.n |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 97 | \( 1 + 15 T + p T^{2} \) | 1.97.p |
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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
−16.56297957280499, −16.08105429061533, −15.55988591504737, −14.88304890584195, −14.52647788145608, −13.84338982560398, −13.46127085067261, −12.61104800781003, −12.09549860643029, −11.74267542020267, −11.16293211470437, −10.40071133126735, −9.781864015961714, −9.194308083627193, −8.344103350864622, −7.745209803767053, −7.183491091242963, −6.461024912953475, −5.804597294040573, −5.243198742899364, −4.209075729602355, −3.935099330446181, −3.162549179701338, −2.169000107592157, −1.392087653229020, 0,
1.392087653229020, 2.169000107592157, 3.162549179701338, 3.935099330446181, 4.209075729602355, 5.243198742899364, 5.804597294040573, 6.461024912953475, 7.183491091242963, 7.745209803767053, 8.344103350864622, 9.194308083627193, 9.781864015961714, 10.40071133126735, 11.16293211470437, 11.74267542020267, 12.09549860643029, 12.61104800781003, 13.46127085067261, 13.84338982560398, 14.52647788145608, 14.88304890584195, 15.55988591504737, 16.08105429061533, 16.56297957280499
