| L(s) = 1 | + 3-s + 2·5-s + 7-s + 9-s − 6·13-s + 2·15-s + 2·17-s + 4·19-s + 21-s + 4·23-s − 25-s + 27-s + 10·29-s + 8·31-s + 2·35-s + 6·37-s − 6·39-s + 2·41-s − 4·43-s + 2·45-s − 8·47-s + 49-s + 2·51-s − 10·53-s + 4·57-s − 12·59-s + 2·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s − 1.66·13-s + 0.516·15-s + 0.485·17-s + 0.917·19-s + 0.218·21-s + 0.834·23-s − 1/5·25-s + 0.192·27-s + 1.85·29-s + 1.43·31-s + 0.338·35-s + 0.986·37-s − 0.960·39-s + 0.312·41-s − 0.609·43-s + 0.298·45-s − 1.16·47-s + 1/7·49-s + 0.280·51-s − 1.37·53-s + 0.529·57-s − 1.56·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.199768350\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.199768350\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 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
−14.60740537471766, −14.18049262127969, −13.87656581222109, −13.37650111912451, −12.63344072270967, −12.28395031376024, −11.70652356308835, −11.13795975958126, −10.27198513329121, −9.943879961710631, −9.600453038219327, −9.058433896717611, −8.309000966121367, −7.755491272053973, −7.448982322757099, −6.489230623966439, −6.285625967687569, −5.234976986397924, −4.877509160710115, −4.457838508231109, −3.269988741269308, −2.863582041465946, −2.270416269957777, −1.493275155867897, −0.7406324998616873,
0.7406324998616873, 1.493275155867897, 2.270416269957777, 2.863582041465946, 3.269988741269308, 4.457838508231109, 4.877509160710115, 5.234976986397924, 6.285625967687569, 6.489230623966439, 7.448982322757099, 7.755491272053973, 8.309000966121367, 9.058433896717611, 9.600453038219327, 9.943879961710631, 10.27198513329121, 11.13795975958126, 11.70652356308835, 12.28395031376024, 12.63344072270967, 13.37650111912451, 13.87656581222109, 14.18049262127969, 14.60740537471766
