| L(s) = 1 | + 3·7-s + 4·11-s + 6·13-s − 6·17-s + 19-s − 6·23-s + 10·29-s − 31-s + 11·37-s + 3·43-s + 8·47-s + 2·49-s − 12·53-s − 6·59-s + 5·61-s + 4·67-s − 2·71-s − 9·73-s + 12·77-s + 5·79-s + 18·83-s − 2·89-s + 18·91-s − 5·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 1.13·7-s + 1.20·11-s + 1.66·13-s − 1.45·17-s + 0.229·19-s − 1.25·23-s + 1.85·29-s − 0.179·31-s + 1.80·37-s + 0.457·43-s + 1.16·47-s + 2/7·49-s − 1.64·53-s − 0.781·59-s + 0.640·61-s + 0.488·67-s − 0.237·71-s − 1.05·73-s + 1.36·77-s + 0.562·79-s + 1.97·83-s − 0.211·89-s + 1.88·91-s − 0.507·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.083352453\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.083352453\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 - 11 T + p T^{2} \) | 1.37.al |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 3 T + p T^{2} \) | 1.43.ad |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 79 | \( 1 - 5 T + p T^{2} \) | 1.79.af |
| 83 | \( 1 - 18 T + p T^{2} \) | 1.83.as |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
| 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
−16.31295571358661, −16.04910004928765, −15.41295102634866, −14.78268442899146, −14.08756615578467, −13.87483037858061, −13.21973676827234, −12.40343450293603, −11.78875816140383, −11.25718824192497, −10.93481892104987, −10.20204273817050, −9.249038641369247, −8.908231591568904, −8.162354331070432, −7.840085155265403, −6.684746240950374, −6.341544757048100, −5.714031110261260, −4.531460183819603, −4.349454042180934, −3.528782242864353, −2.436863577005057, −1.592227899926598, −0.8982115126896610,
0.8982115126896610, 1.592227899926598, 2.436863577005057, 3.528782242864353, 4.349454042180934, 4.531460183819603, 5.714031110261260, 6.341544757048100, 6.684746240950374, 7.840085155265403, 8.162354331070432, 8.908231591568904, 9.249038641369247, 10.20204273817050, 10.93481892104987, 11.25718824192497, 11.78875816140383, 12.40343450293603, 13.21973676827234, 13.87483037858061, 14.08756615578467, 14.78268442899146, 15.41295102634866, 16.04910004928765, 16.31295571358661
