| L(s) = 1 | − 2·5-s − 11-s + 6·13-s + 5·17-s − 7·19-s − 4·23-s − 25-s − 4·29-s − 6·31-s + 2·37-s − 3·41-s + 43-s + 12·53-s + 2·55-s − 7·59-s + 12·61-s − 12·65-s − 13·67-s + 8·71-s − 73-s + 6·79-s + 16·83-s − 10·85-s + 6·89-s + 14·95-s + 5·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.301·11-s + 1.66·13-s + 1.21·17-s − 1.60·19-s − 0.834·23-s − 1/5·25-s − 0.742·29-s − 1.07·31-s + 0.328·37-s − 0.468·41-s + 0.152·43-s + 1.64·53-s + 0.269·55-s − 0.911·59-s + 1.53·61-s − 1.48·65-s − 1.58·67-s + 0.949·71-s − 0.117·73-s + 0.675·79-s + 1.75·83-s − 1.08·85-s + 0.635·89-s + 1.43·95-s + 0.507·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 7 T + p T^{2} \) | 1.59.h |
| 61 | \( 1 - 12 T + p T^{2} \) | 1.61.am |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + T + p T^{2} \) | 1.73.b |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
| 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
−14.70442383463072, −13.94097062437263, −13.44425389078217, −13.00729836918579, −12.46249096847408, −11.98064520586613, −11.42257414992319, −11.03614880510397, −10.41625493522676, −10.15941063092907, −9.211126788235655, −8.817511333296212, −8.252114529017717, −7.804736601211768, −7.431588226632852, −6.575972948459634, −6.105059405883603, −5.617220073530686, −4.946506441935722, −4.022415960739349, −3.825567231284924, −3.361021352787522, −2.339053847427406, −1.725688685518272, −0.8394139306323873, 0,
0.8394139306323873, 1.725688685518272, 2.339053847427406, 3.361021352787522, 3.825567231284924, 4.022415960739349, 4.946506441935722, 5.617220073530686, 6.105059405883603, 6.575972948459634, 7.431588226632852, 7.804736601211768, 8.252114529017717, 8.817511333296212, 9.211126788235655, 10.15941063092907, 10.41625493522676, 11.03614880510397, 11.42257414992319, 11.98064520586613, 12.46249096847408, 13.00729836918579, 13.44425389078217, 13.94097062437263, 14.70442383463072
