| L(s) = 1 | − 5-s + 2·7-s + 3·11-s + 13-s − 3·17-s − 8·19-s + 3·23-s + 25-s + 9·29-s − 7·31-s − 2·35-s − 2·37-s − 12·41-s + 7·43-s − 3·47-s − 3·49-s − 12·53-s − 3·55-s − 12·59-s + 10·61-s − 65-s + 4·67-s + 2·73-s + 6·77-s − 79-s − 18·83-s + 3·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.755·7-s + 0.904·11-s + 0.277·13-s − 0.727·17-s − 1.83·19-s + 0.625·23-s + 1/5·25-s + 1.67·29-s − 1.25·31-s − 0.338·35-s − 0.328·37-s − 1.87·41-s + 1.06·43-s − 0.437·47-s − 3/7·49-s − 1.64·53-s − 0.404·55-s − 1.56·59-s + 1.28·61-s − 0.124·65-s + 0.488·67-s + 0.234·73-s + 0.683·77-s − 0.112·79-s − 1.97·83-s + 0.325·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 + 18 T + p T^{2} \) | 1.83.s |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
| 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
−7.43430060198272589745088294377, −6.55524367608411081354569325404, −6.38697348080489145122745720268, −5.16010405009510092024867246227, −4.57294598374498678616501508540, −3.99620794878226514413680665855, −3.14550759871192573821978902336, −2.07466326223306564920409737248, −1.33827976943654901646256470832, 0,
1.33827976943654901646256470832, 2.07466326223306564920409737248, 3.14550759871192573821978902336, 3.99620794878226514413680665855, 4.57294598374498678616501508540, 5.16010405009510092024867246227, 6.38697348080489145122745720268, 6.55524367608411081354569325404, 7.43430060198272589745088294377
