| L(s) = 1 | + 3·3-s + 2·5-s − 2·7-s + 6·9-s − 11-s + 7·13-s + 6·15-s + 7·19-s − 6·21-s − 6·23-s − 25-s + 9·27-s + 3·29-s + 2·31-s − 3·33-s − 4·35-s + 6·37-s + 21·39-s − 6·41-s − 7·43-s + 12·45-s − 8·47-s − 3·49-s − 6·53-s − 2·55-s + 21·57-s + 14·59-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.894·5-s − 0.755·7-s + 2·9-s − 0.301·11-s + 1.94·13-s + 1.54·15-s + 1.60·19-s − 1.30·21-s − 1.25·23-s − 1/5·25-s + 1.73·27-s + 0.557·29-s + 0.359·31-s − 0.522·33-s − 0.676·35-s + 0.986·37-s + 3.36·39-s − 0.937·41-s − 1.06·43-s + 1.78·45-s − 1.16·47-s − 3/7·49-s − 0.824·53-s − 0.269·55-s + 2.78·57-s + 1.82·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 383792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 383792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.025958743\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.025958743\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 17 | \( 1 \) | |
| 83 | \( 1 - T \) | |
| good | 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 - 7 T + p T^{2} \) | 1.13.ah |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 + 15 T + p T^{2} \) | 1.61.p |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 + T + p T^{2} \) | 1.73.b |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 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
−12.79704882946515, −12.03802953667157, −11.61957392102384, −11.08537120605732, −10.33732008583468, −10.05967893873203, −9.555461064502619, −9.515681787586843, −8.786228211375522, −8.421662747958489, −7.919143871583044, −7.774100268197916, −6.881039439251348, −6.463610920793334, −6.164238853527821, −5.514155476288039, −5.000327418211123, −4.224786910419382, −3.704127531921953, −3.347785084245027, −2.958433329892097, −2.412895592705980, −1.581534987696501, −1.551135736504678, −0.6117963335793934,
0.6117963335793934, 1.551135736504678, 1.581534987696501, 2.412895592705980, 2.958433329892097, 3.347785084245027, 3.704127531921953, 4.224786910419382, 5.000327418211123, 5.514155476288039, 6.164238853527821, 6.463610920793334, 6.881039439251348, 7.774100268197916, 7.919143871583044, 8.421662747958489, 8.786228211375522, 9.515681787586843, 9.555461064502619, 10.05967893873203, 10.33732008583468, 11.08537120605732, 11.61957392102384, 12.03802953667157, 12.79704882946515
