| L(s) = 1 | + 6·11-s + 3·13-s − 23-s − 5·25-s + 3·29-s + 7·31-s + 8·37-s − 11·41-s + 4·43-s + 47-s − 4·53-s + 12·59-s + 6·61-s + 12·67-s + 5·71-s − 15·73-s − 4·79-s − 12·89-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 1.80·11-s + 0.832·13-s − 0.208·23-s − 25-s + 0.557·29-s + 1.25·31-s + 1.31·37-s − 1.71·41-s + 0.609·43-s + 0.145·47-s − 0.549·53-s + 1.56·59-s + 0.768·61-s + 1.46·67-s + 0.593·71-s − 1.75·73-s − 0.450·79-s − 1.27·89-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-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
| \(L(1)\) |
\(\approx\) |
\(3.405408937\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.405408937\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 11 T + p T^{2} \) | 1.41.l |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - T + p T^{2} \) | 1.47.ab |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 + 15 T + p T^{2} \) | 1.73.p |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 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
−14.09625859506087, −13.43495505882593, −13.16258678822286, −12.35788214771777, −11.92315791989842, −11.51258136150076, −11.20836404798221, −10.44159361701251, −9.737853394809113, −9.689299842813092, −8.848010946002262, −8.430892754918249, −8.065515304113941, −7.230156352872471, −6.675930575527607, −6.336330744559869, −5.815269488136689, −5.151768252435075, −4.242028363554715, −4.093564805052486, −3.428822966878049, −2.717823978520966, −1.895744801180052, −1.265341497154439, −0.6585699314353960,
0.6585699314353960, 1.265341497154439, 1.895744801180052, 2.717823978520966, 3.428822966878049, 4.093564805052486, 4.242028363554715, 5.151768252435075, 5.815269488136689, 6.336330744559869, 6.675930575527607, 7.230156352872471, 8.065515304113941, 8.430892754918249, 8.848010946002262, 9.689299842813092, 9.737853394809113, 10.44159361701251, 11.20836404798221, 11.51258136150076, 11.92315791989842, 12.35788214771777, 13.16258678822286, 13.43495505882593, 14.09625859506087
