| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 8-s + 9-s − 10-s + 11-s − 12-s + 2·13-s − 15-s + 16-s − 2·17-s − 18-s + 4·19-s + 20-s − 22-s + 24-s + 25-s − 2·26-s − 27-s + 2·29-s + 30-s − 32-s − 33-s + 2·34-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 + 0.301·11-s − 0.288·12-s + 0.554·13-s − 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.917·19-s + 0.223·20-s − 0.213·22-s + 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.371·29-s + 0.182·30-s − 0.176·32-s − 0.174·33-s + 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 317130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 317130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.872388325\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.872388325\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| 31 | \( 1 \) | |
| good | 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.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 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
−12.73439662111227, −11.93967661004091, −11.64507454755118, −11.23818808158273, −10.86285013257490, −10.26737749222817, −9.939947367954292, −9.416049823621325, −9.099846219197633, −8.549219886159912, −7.998305765552108, −7.614664409578998, −6.948641156231087, −6.622341395618616, −6.133152985518407, −5.679852615629450, −5.223066394532983, −4.533575445450018, −4.116642710527132, −3.342036591007479, −2.878712702662908, −2.184040557538639, −1.583879824798884, −1.035880907975690, −0.4832786834389820,
0.4832786834389820, 1.035880907975690, 1.583879824798884, 2.184040557538639, 2.878712702662908, 3.342036591007479, 4.116642710527132, 4.533575445450018, 5.223066394532983, 5.679852615629450, 6.133152985518407, 6.622341395618616, 6.948641156231087, 7.614664409578998, 7.998305765552108, 8.549219886159912, 9.099846219197633, 9.416049823621325, 9.939947367954292, 10.26737749222817, 10.86285013257490, 11.23818808158273, 11.64507454755118, 11.93967661004091, 12.73439662111227
