| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 2·7-s − 8-s + 9-s − 10-s − 11-s − 12-s + 2·14-s − 15-s + 16-s − 3·17-s − 18-s − 19-s + 20-s + 2·21-s + 22-s + 4·23-s + 24-s + 25-s − 27-s − 2·28-s + 10·29-s + 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.301·11-s − 0.288·12-s + 0.534·14-s − 0.258·15-s + 1/4·16-s − 0.727·17-s − 0.235·18-s − 0.229·19-s + 0.223·20-s + 0.436·21-s + 0.213·22-s + 0.834·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s − 0.377·28-s + 1.85·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 96330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 + 11 T + p T^{2} \) | 1.53.l |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + T + p T^{2} \) | 1.71.b |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 + 5 T + p T^{2} \) | 1.79.f |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 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
−13.82399067675844, −13.62507762980705, −12.86952522341248, −12.60128869853457, −12.05092173590850, −11.53322250893207, −10.90852652642129, −10.54787838705378, −10.13437547972538, −9.653206502103126, −9.044977680636468, −8.731952112115492, −8.072310975794860, −7.452493368036386, −6.876053947277338, −6.379327173583864, −6.199527358732071, −5.387077377959303, −4.789849702976846, −4.355278325344057, −3.297470461872391, −2.952022569888046, −2.211466283559419, −1.498653352887357, −0.7462564137357783, 0,
0.7462564137357783, 1.498653352887357, 2.211466283559419, 2.952022569888046, 3.297470461872391, 4.355278325344057, 4.789849702976846, 5.387077377959303, 6.199527358732071, 6.379327173583864, 6.876053947277338, 7.452493368036386, 8.072310975794860, 8.731952112115492, 9.044977680636468, 9.653206502103126, 10.13437547972538, 10.54787838705378, 10.90852652642129, 11.53322250893207, 12.05092173590850, 12.60128869853457, 12.86952522341248, 13.62507762980705, 13.82399067675844
