| L(s) = 1 | + 4·5-s + 7-s + 11-s − 6·13-s + 2·19-s − 4·23-s + 11·25-s − 2·29-s − 2·31-s + 4·35-s − 2·37-s + 4·43-s − 6·47-s + 49-s + 2·53-s + 4·55-s − 14·61-s − 24·65-s + 12·67-s − 8·71-s − 4·73-s + 77-s + 8·79-s + 2·83-s + 14·89-s − 6·91-s + 8·95-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 0.377·7-s + 0.301·11-s − 1.66·13-s + 0.458·19-s − 0.834·23-s + 11/5·25-s − 0.371·29-s − 0.359·31-s + 0.676·35-s − 0.328·37-s + 0.609·43-s − 0.875·47-s + 1/7·49-s + 0.274·53-s + 0.539·55-s − 1.79·61-s − 2.97·65-s + 1.46·67-s − 0.949·71-s − 0.468·73-s + 0.113·77-s + 0.900·79-s + 0.219·83-s + 1.48·89-s − 0.628·91-s + 0.820·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.298312879\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.298312879\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
| 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
−14.65505183315709, −14.10429486253587, −13.75759942471364, −13.17339370950705, −12.64953476560878, −12.10569312152067, −11.68810630563918, −10.80885452908971, −10.45998817276524, −9.770514174760455, −9.564792776024709, −9.083133465333684, −8.379288008176381, −7.622183773900114, −7.189921252275511, −6.506241168647081, −5.955751374849411, −5.440381349850976, −4.922574289073444, −4.409633199342604, −3.386987553330019, −2.693940184611027, −1.994582718539007, −1.713427539662560, −0.6231211516248812,
0.6231211516248812, 1.713427539662560, 1.994582718539007, 2.693940184611027, 3.386987553330019, 4.409633199342604, 4.922574289073444, 5.440381349850976, 5.955751374849411, 6.506241168647081, 7.189921252275511, 7.622183773900114, 8.379288008176381, 9.083133465333684, 9.564792776024709, 9.770514174760455, 10.45998817276524, 10.80885452908971, 11.68810630563918, 12.10569312152067, 12.64953476560878, 13.17339370950705, 13.75759942471364, 14.10429486253587, 14.65505183315709
