| L(s) = 1 | + 2·2-s + 3-s + 2·4-s + 2·6-s + 7-s + 9-s + 6·11-s + 2·12-s − 13-s + 2·14-s − 4·16-s + 2·18-s − 4·19-s + 21-s + 12·22-s − 6·23-s − 2·26-s + 27-s + 2·28-s − 10·29-s − 4·31-s − 8·32-s + 6·33-s + 2·36-s − 7·37-s − 8·38-s − 39-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s + 0.816·6-s + 0.377·7-s + 1/3·9-s + 1.80·11-s + 0.577·12-s − 0.277·13-s + 0.534·14-s − 16-s + 0.471·18-s − 0.917·19-s + 0.218·21-s + 2.55·22-s − 1.25·23-s − 0.392·26-s + 0.192·27-s + 0.377·28-s − 1.85·29-s − 0.718·31-s − 1.41·32-s + 1.04·33-s + 1/3·36-s − 1.15·37-s − 1.29·38-s − 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 281775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 281775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.289374275\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.289374275\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 2 | \( 1 - p T + p T^{2} \) | 1.2.ac |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 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.ae |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−12.88067279604642, −12.39560938349015, −11.92704103904420, −11.46351750563101, −11.28461129656470, −10.51190869912246, −9.992148420550756, −9.431442793742514, −8.955929407518493, −8.731686816471434, −8.038867572192783, −7.480230465529395, −6.896854031328726, −6.619581948564515, −5.987653356350787, −5.623355005282105, −4.988169696804751, −4.480473763594052, −3.880176747802712, −3.738598729188355, −3.322243804432875, −2.257342947413820, −2.063379007116064, −1.519932464653816, −0.4625285168833327,
0.4625285168833327, 1.519932464653816, 2.063379007116064, 2.257342947413820, 3.322243804432875, 3.738598729188355, 3.880176747802712, 4.480473763594052, 4.988169696804751, 5.623355005282105, 5.987653356350787, 6.619581948564515, 6.896854031328726, 7.480230465529395, 8.038867572192783, 8.731686816471434, 8.955929407518493, 9.431442793742514, 9.992148420550756, 10.51190869912246, 11.28461129656470, 11.46351750563101, 11.92704103904420, 12.39560938349015, 12.88067279604642
