| L(s) = 1 | − 7-s − 6·11-s + 5·13-s + 6·17-s + 5·19-s + 6·23-s + 6·29-s − 31-s + 2·37-s − 43-s − 6·47-s − 6·49-s + 12·53-s + 6·59-s − 13·61-s + 11·67-s + 2·73-s + 6·77-s + 8·79-s + 6·83-s − 5·91-s − 7·97-s + 12·101-s − 4·103-s − 12·107-s − 7·109-s − 12·113-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 1.80·11-s + 1.38·13-s + 1.45·17-s + 1.14·19-s + 1.25·23-s + 1.11·29-s − 0.179·31-s + 0.328·37-s − 0.152·43-s − 0.875·47-s − 6/7·49-s + 1.64·53-s + 0.781·59-s − 1.66·61-s + 1.34·67-s + 0.234·73-s + 0.683·77-s + 0.900·79-s + 0.658·83-s − 0.524·91-s − 0.710·97-s + 1.19·101-s − 0.394·103-s − 1.16·107-s − 0.670·109-s − 1.12·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.532714580\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.532714580\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
| 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
−10.19494219048266918540495603294, −9.350774149991784768341322845696, −8.275919068288280643535776826565, −7.72957625844481983843839527768, −6.68901441807918237505792173786, −5.61334343701912201622834145502, −5.03310337383134036338192767807, −3.50841194957311191464253456169, −2.81121112269996318424524765777, −1.03773521644426615440225296994,
1.03773521644426615440225296994, 2.81121112269996318424524765777, 3.50841194957311191464253456169, 5.03310337383134036338192767807, 5.61334343701912201622834145502, 6.68901441807918237505792173786, 7.72957625844481983843839527768, 8.275919068288280643535776826565, 9.350774149991784768341322845696, 10.19494219048266918540495603294
