| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s + 4·7-s − 8-s + 9-s − 10-s + 4·11-s + 12-s + 6·13-s − 4·14-s + 15-s + 16-s + 17-s − 18-s − 19-s + 20-s + 4·21-s − 4·22-s − 8·23-s − 24-s + 25-s − 6·26-s + 27-s + 4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s + 0.288·12-s + 1.66·13-s − 1.06·14-s + 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.229·19-s + 0.223·20-s + 0.872·21-s − 0.852·22-s − 1.66·23-s − 0.204·24-s + 1/5·25-s − 1.17·26-s + 0.192·27-s + 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.209170685\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.209170685\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 17 | \( 1 - T \) | |
| 19 | \( 1 + T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−7.902887167306318609373951967262, −7.15426630059697780001857818820, −6.35621150548566070941818133119, −5.84181661123687240591548233243, −4.91801609441937171125607459099, −3.97025300809406719573230962063, −3.53766564618941878390716102170, −2.18716422953651562952483405123, −1.67299374913606626375318494086, −1.03244227461781561340317132303,
1.03244227461781561340317132303, 1.67299374913606626375318494086, 2.18716422953651562952483405123, 3.53766564618941878390716102170, 3.97025300809406719573230962063, 4.91801609441937171125607459099, 5.84181661123687240591548233243, 6.35621150548566070941818133119, 7.15426630059697780001857818820, 7.902887167306318609373951967262
