| L(s) = 1 | + 2-s − 3-s + 4-s + 3·5-s − 6-s − 7-s + 8-s + 9-s + 3·10-s + 11-s − 12-s − 13-s − 14-s − 3·15-s + 16-s + 7·17-s + 18-s + 19-s + 3·20-s + 21-s + 22-s − 7·23-s − 24-s + 4·25-s − 26-s − 27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.34·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.948·10-s + 0.301·11-s − 0.288·12-s − 0.277·13-s − 0.267·14-s − 0.774·15-s + 1/4·16-s + 1.69·17-s + 0.235·18-s + 0.229·19-s + 0.670·20-s + 0.218·21-s + 0.213·22-s − 1.45·23-s − 0.204·24-s + 4/5·25-s − 0.196·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.278357824\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.278357824\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 7 T + p T^{2} \) | 1.23.h |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 + 6 T + p T^{2} \) | 1.67.g |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 5 T + p T^{2} \) | 1.73.af |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−10.72405745965352481313791757742, −9.988795367837157057695788102202, −9.435514649834190793278551180591, −7.925758442786961590691428846639, −6.84627164270088746798526670414, −5.86366672737057092676215077124, −5.55921197766306254502353102551, −4.23861623857583401252172064502, −2.89830493976033687883443576173, −1.52673365627799806411662380976,
1.52673365627799806411662380976, 2.89830493976033687883443576173, 4.23861623857583401252172064502, 5.55921197766306254502353102551, 5.86366672737057092676215077124, 6.84627164270088746798526670414, 7.925758442786961590691428846639, 9.435514649834190793278551180591, 9.988795367837157057695788102202, 10.72405745965352481313791757742
