| L(s) = 1 | − 2·5-s + 3·11-s + 2·13-s + 7·17-s + 4·19-s + 23-s − 25-s − 9·29-s − 10·31-s + 6·37-s + 2·41-s − 2·43-s − 3·47-s − 12·53-s − 6·55-s + 12·59-s + 10·61-s − 4·65-s + 8·67-s + 3·71-s + 11·73-s + 79-s − 12·83-s − 14·85-s + 6·89-s − 8·95-s − 10·97-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.904·11-s + 0.554·13-s + 1.69·17-s + 0.917·19-s + 0.208·23-s − 1/5·25-s − 1.67·29-s − 1.79·31-s + 0.986·37-s + 0.312·41-s − 0.304·43-s − 0.437·47-s − 1.64·53-s − 0.809·55-s + 1.56·59-s + 1.28·61-s − 0.496·65-s + 0.977·67-s + 0.356·71-s + 1.28·73-s + 0.112·79-s − 1.31·83-s − 1.51·85-s + 0.635·89-s − 0.820·95-s − 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 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.am |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
| 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
−14.31698653631960, −13.85213860750123, −13.10966464339438, −12.66544243909460, −12.31254323620009, −11.51596509285965, −11.36453148832832, −11.04770957767626, −10.12373638288592, −9.564895989075905, −9.385999843257872, −8.631862525290959, −7.962565551861338, −7.733190514410601, −7.146690253981617, −6.647309112891930, −5.793384207005669, −5.514534851590226, −4.884098649556395, −3.870837272084396, −3.731327248887902, −3.335027550388412, −2.344148357048550, −1.471656558295794, −0.9797774305208540, 0,
0.9797774305208540, 1.471656558295794, 2.344148357048550, 3.335027550388412, 3.731327248887902, 3.870837272084396, 4.884098649556395, 5.514534851590226, 5.793384207005669, 6.647309112891930, 7.146690253981617, 7.733190514410601, 7.962565551861338, 8.631862525290959, 9.385999843257872, 9.564895989075905, 10.12373638288592, 11.04770957767626, 11.36453148832832, 11.51596509285965, 12.31254323620009, 12.66544243909460, 13.10966464339438, 13.85213860750123, 14.31698653631960
