| L(s) = 1 | − 2-s − 3-s − 4-s − 5-s + 6-s − 4·7-s + 3·8-s + 9-s + 10-s − 4·11-s + 12-s + 13-s + 4·14-s + 15-s − 16-s − 17-s − 18-s − 4·19-s + 20-s + 4·21-s + 4·22-s − 3·24-s + 25-s − 26-s − 27-s + 4·28-s − 6·29-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 + 1.06·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s + 0.288·12-s + 0.277·13-s + 1.06·14-s + 0.258·15-s − 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.917·19-s + 0.223·20-s + 0.872·21-s + 0.852·22-s − 0.612·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s + 0.755·28-s − 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| 17 | \( 1 + T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−8.089648190244173196108810762220, −7.06459084579843194945265993639, −6.63034994304785477402170173649, −5.55441534351708659869170985489, −4.90555900723098597346773746285, −3.88506522825871946059410431760, −3.17177515385687276813064931079, −1.76656905123940013148236099690, 0, 0,
1.76656905123940013148236099690, 3.17177515385687276813064931079, 3.88506522825871946059410431760, 4.90555900723098597346773746285, 5.55441534351708659869170985489, 6.63034994304785477402170173649, 7.06459084579843194945265993639, 8.089648190244173196108810762220
