| L(s) = 1 | − 2-s + 4-s − 8-s − 3·9-s − 11-s + 2·13-s + 16-s + 2·17-s + 3·18-s + 22-s + 4·23-s − 2·26-s − 6·29-s − 8·31-s − 32-s − 2·34-s − 3·36-s + 6·37-s − 2·41-s − 4·43-s − 44-s − 4·46-s − 12·47-s − 7·49-s + 2·52-s − 10·53-s + 6·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 9-s − 0.301·11-s + 0.554·13-s + 1/4·16-s + 0.485·17-s + 0.707·18-s + 0.213·22-s + 0.834·23-s − 0.392·26-s − 1.11·29-s − 1.43·31-s − 0.176·32-s − 0.342·34-s − 1/2·36-s + 0.986·37-s − 0.312·41-s − 0.609·43-s − 0.150·44-s − 0.589·46-s − 1.75·47-s − 49-s + 0.277·52-s − 1.37·53-s + 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| 19 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.31170647759630, −13.03909100131941, −12.75118721588352, −11.85629866140620, −11.60338652989933, −11.06293126056202, −10.88940390904666, −10.24491448428929, −9.610636174369771, −9.327777874956897, −8.832271566959421, −8.294859693226594, −7.899759238889167, −7.514448460983957, −6.771196955139177, −6.439158398939614, −5.743857804667175, −5.421388096596447, −4.888940285180026, −4.078066458732183, −3.386521067504343, −3.064933428919448, −2.461695183282345, −1.589885038418505, −1.298503621771160, 0, 0,
1.298503621771160, 1.589885038418505, 2.461695183282345, 3.064933428919448, 3.386521067504343, 4.078066458732183, 4.888940285180026, 5.421388096596447, 5.743857804667175, 6.439158398939614, 6.771196955139177, 7.514448460983957, 7.899759238889167, 8.294859693226594, 8.832271566959421, 9.327777874956897, 9.610636174369771, 10.24491448428929, 10.88940390904666, 11.06293126056202, 11.60338652989933, 11.85629866140620, 12.75118721588352, 13.03909100131941, 13.31170647759630
