| L(s) = 1 | + 5-s + 4·7-s + 13-s − 17-s + 7·23-s + 25-s − 10·29-s + 3·31-s + 4·35-s + 12·37-s + 2·41-s + 10·43-s − 3·47-s + 9·49-s − 5·53-s + 14·59-s + 5·61-s + 65-s + 12·67-s + 8·71-s + 16·73-s + 3·79-s − 8·83-s − 85-s + 4·91-s + 12·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.51·7-s + 0.277·13-s − 0.242·17-s + 1.45·23-s + 1/5·25-s − 1.85·29-s + 0.538·31-s + 0.676·35-s + 1.97·37-s + 0.312·41-s + 1.52·43-s − 0.437·47-s + 9/7·49-s − 0.686·53-s + 1.82·59-s + 0.640·61-s + 0.124·65-s + 1.46·67-s + 0.949·71-s + 1.87·73-s + 0.337·79-s − 0.878·83-s − 0.108·85-s + 0.419·91-s + 1.21·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 283140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.548013525\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.548013525\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 7 T + p T^{2} \) | 1.23.ah |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 - 12 T + p T^{2} \) | 1.37.am |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 5 T + p T^{2} \) | 1.53.f |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 16 T + p T^{2} \) | 1.73.aq |
| 79 | \( 1 - 3 T + p T^{2} \) | 1.79.ad |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.85948841041407, −12.35692368490031, −11.52572724509187, −11.35770861163502, −10.97894003722951, −10.69102027494298, −9.894891541373227, −9.363109337783002, −9.217420623990109, −8.474861222892765, −8.057553712646077, −7.755442576254906, −7.087294593863368, −6.722271489591559, −6.015026926542963, −5.541405180055398, −5.141745819541430, −4.659047335179717, −4.106537512698744, −3.627371498397292, −2.807213794869684, −2.231177736761809, −1.890811280844047, −0.9940605036153031, −0.7635806006137174,
0.7635806006137174, 0.9940605036153031, 1.890811280844047, 2.231177736761809, 2.807213794869684, 3.627371498397292, 4.106537512698744, 4.659047335179717, 5.141745819541430, 5.541405180055398, 6.015026926542963, 6.722271489591559, 7.087294593863368, 7.755442576254906, 8.057553712646077, 8.474861222892765, 9.217420623990109, 9.363109337783002, 9.894891541373227, 10.69102027494298, 10.97894003722951, 11.35770861163502, 11.52572724509187, 12.35692368490031, 12.85948841041407
