| L(s) = 1 | − 2-s + 4-s − 8-s − 6·11-s + 2·13-s + 16-s − 2·17-s + 6·22-s − 23-s − 5·25-s − 2·26-s + 2·29-s + 2·31-s − 32-s + 2·34-s − 10·37-s − 6·41-s + 4·43-s − 6·44-s + 46-s + 4·47-s − 7·49-s + 5·50-s + 2·52-s − 10·53-s − 2·58-s − 4·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1.80·11-s + 0.554·13-s + 1/4·16-s − 0.485·17-s + 1.27·22-s − 0.208·23-s − 25-s − 0.392·26-s + 0.371·29-s + 0.359·31-s − 0.176·32-s + 0.342·34-s − 1.64·37-s − 0.937·41-s + 0.609·43-s − 0.904·44-s + 0.147·46-s + 0.583·47-s − 49-s + 0.707·50-s + 0.277·52-s − 1.37·53-s − 0.262·58-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 149454 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149454 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 19 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 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.g |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 - 10 T + p T^{2} \) | 1.83.ak |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−13.92099941403549, −13.31528916550292, −12.86784273952342, −12.33802173806470, −11.91020297450108, −11.27355298433729, −10.80038452260563, −10.54152503537316, −9.983690825528542, −9.587915210921256, −8.935710007869246, −8.436029672417668, −8.035064739126618, −7.673831875078060, −7.059905409733508, −6.531656955867428, −5.958851774181117, −5.468781850367179, −4.915913453816375, −4.357025942093197, −3.523791619499304, −3.059770911576320, −2.416274027298992, −1.866579931278528, −1.209214079976432, 0, 0,
1.209214079976432, 1.866579931278528, 2.416274027298992, 3.059770911576320, 3.523791619499304, 4.357025942093197, 4.915913453816375, 5.468781850367179, 5.958851774181117, 6.531656955867428, 7.059905409733508, 7.673831875078060, 8.035064739126618, 8.436029672417668, 8.935710007869246, 9.587915210921256, 9.983690825528542, 10.54152503537316, 10.80038452260563, 11.27355298433729, 11.91020297450108, 12.33802173806470, 12.86784273952342, 13.31528916550292, 13.92099941403549
