| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s − 4·11-s + 12-s + 13-s − 15-s + 16-s + 2·17-s − 18-s + 19-s − 20-s + 4·22-s + 4·23-s − 24-s + 25-s − 26-s + 27-s + 6·29-s + 30-s − 4·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s + 0.288·12-s + 0.277·13-s − 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.229·19-s − 0.223·20-s + 0.852·22-s + 0.834·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s + 1.11·29-s + 0.182·30-s − 0.718·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363090 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363090 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| 19 | \( 1 - T \) | |
| good | 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 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.ag |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 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 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−12.76559228567260, −12.21393856235015, −11.87777721450700, −11.24698230588913, −10.78535283889126, −10.52301145435430, −10.00117059880699, −9.496610243591394, −9.158541099919286, −8.404948848123920, −8.284066588272737, −7.873458686748213, −7.330031348659955, −6.848263433139707, −6.554795575057198, −5.634925446374330, −5.352674342116525, −4.752866654170404, −4.205492590421007, −3.456321534868435, −3.024750031055742, −2.771876094202189, −1.891450138747197, −1.459270941810730, −0.6918866281500920, 0,
0.6918866281500920, 1.459270941810730, 1.891450138747197, 2.771876094202189, 3.024750031055742, 3.456321534868435, 4.205492590421007, 4.752866654170404, 5.352674342116525, 5.634925446374330, 6.554795575057198, 6.848263433139707, 7.330031348659955, 7.873458686748213, 8.284066588272737, 8.404948848123920, 9.158541099919286, 9.496610243591394, 10.00117059880699, 10.52301145435430, 10.78535283889126, 11.24698230588913, 11.87777721450700, 12.21393856235015, 12.76559228567260
