| L(s) = 1 | + 2-s + 4-s + 5-s + 7-s + 8-s + 10-s + 4·13-s + 14-s + 16-s + 19-s + 20-s − 4·23-s − 4·25-s + 4·26-s + 28-s + 8·29-s + 32-s + 35-s + 10·37-s + 38-s + 40-s − 4·46-s − 47-s − 6·49-s − 4·50-s + 4·52-s − 12·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s + 0.353·8-s + 0.316·10-s + 1.10·13-s + 0.267·14-s + 1/4·16-s + 0.229·19-s + 0.223·20-s − 0.834·23-s − 4/5·25-s + 0.784·26-s + 0.188·28-s + 1.48·29-s + 0.176·32-s + 0.169·35-s + 1.64·37-s + 0.162·38-s + 0.158·40-s − 0.589·46-s − 0.145·47-s − 6/7·49-s − 0.565·50-s + 0.554·52-s − 1.64·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 272322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 272322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 41 | \( 1 \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 7 T + p T^{2} \) | 1.71.h |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 - 5 T + p T^{2} \) | 1.79.af |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 97 | \( 1 + 4 T + p T^{2} \) | 1.97.e |
| 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
−13.17356896702948, −12.46697068534707, −12.23746396093549, −11.58840484763149, −11.17644168969337, −10.95217601933125, −10.19397538453451, −9.886742063454728, −9.412951378468743, −8.761320403797022, −8.236621866683742, −7.890760238350985, −7.431396847180445, −6.627869975708944, −6.256128092229805, −5.975537897341392, −5.429981846119124, −4.783829459718010, −4.354274595988870, −3.934657139816572, −3.170268970236082, −2.836153499867248, −2.085945105783093, −1.510208672334893, −1.065810817303497, 0,
1.065810817303497, 1.510208672334893, 2.085945105783093, 2.836153499867248, 3.170268970236082, 3.934657139816572, 4.354274595988870, 4.783829459718010, 5.429981846119124, 5.975537897341392, 6.256128092229805, 6.627869975708944, 7.431396847180445, 7.890760238350985, 8.236621866683742, 8.761320403797022, 9.412951378468743, 9.886742063454728, 10.19397538453451, 10.95217601933125, 11.17644168969337, 11.58840484763149, 12.23746396093549, 12.46697068534707, 13.17356896702948
