| L(s) = 1 | − 2-s + 4-s + 5-s + 4·7-s − 8-s − 10-s − 2·13-s − 4·14-s + 16-s + 6·17-s + 4·19-s + 20-s + 25-s + 2·26-s + 4·28-s − 6·29-s + 8·31-s − 32-s − 6·34-s + 4·35-s + 2·37-s − 4·38-s − 40-s − 6·41-s + 4·43-s + 9·49-s − 50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 1.51·7-s − 0.353·8-s − 0.316·10-s − 0.554·13-s − 1.06·14-s + 1/4·16-s + 1.45·17-s + 0.917·19-s + 0.223·20-s + 1/5·25-s + 0.392·26-s + 0.755·28-s − 1.11·29-s + 1.43·31-s − 0.176·32-s − 1.02·34-s + 0.676·35-s + 0.328·37-s − 0.648·38-s − 0.158·40-s − 0.937·41-s + 0.609·43-s + 9/7·49-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.333997183\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.333997183\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 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
−16.69034153336038, −16.12411205350818, −15.24174808221486, −14.90592613662044, −14.19200640944340, −13.94991512828057, −13.05122282598828, −12.29303208078547, −11.65614098761563, −11.45569267426119, −10.55440214701818, −10.03011761755423, −9.600371326177063, −8.765418172148747, −8.228646632892931, −7.571067501960620, −7.282001995403109, −6.250542754906307, −5.441726033680890, −5.122083261442397, −4.182674130816111, −3.184297288240087, −2.348790039617709, −1.538129754228387, −0.8644374413898091,
0.8644374413898091, 1.538129754228387, 2.348790039617709, 3.184297288240087, 4.182674130816111, 5.122083261442397, 5.441726033680890, 6.250542754906307, 7.282001995403109, 7.571067501960620, 8.228646632892931, 8.765418172148747, 9.600371326177063, 10.03011761755423, 10.55440214701818, 11.45569267426119, 11.65614098761563, 12.29303208078547, 13.05122282598828, 13.94991512828057, 14.19200640944340, 14.90592613662044, 15.24174808221486, 16.12411205350818, 16.69034153336038
