| L(s) = 1 | − 3·5-s − 7-s − 3·11-s − 13-s − 6·17-s − 4·19-s + 3·23-s + 4·25-s − 3·29-s + 5·31-s + 3·35-s + 2·37-s − 3·41-s − 43-s + 9·47-s − 6·49-s + 6·53-s + 9·55-s + 3·59-s − 13·61-s + 3·65-s − 7·67-s + 12·71-s − 10·73-s + 3·77-s + 11·79-s + 9·83-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 0.377·7-s − 0.904·11-s − 0.277·13-s − 1.45·17-s − 0.917·19-s + 0.625·23-s + 4/5·25-s − 0.557·29-s + 0.898·31-s + 0.507·35-s + 0.328·37-s − 0.468·41-s − 0.152·43-s + 1.31·47-s − 6/7·49-s + 0.824·53-s + 1.21·55-s + 0.390·59-s − 1.66·61-s + 0.372·65-s − 0.855·67-s + 1.42·71-s − 1.17·73-s + 0.341·77-s + 1.23·79-s + 0.987·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 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.b |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 + 7 T + p T^{2} \) | 1.67.h |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 11 T + p T^{2} \) | 1.97.al |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.13154017588271625556961179667, −10.45530561787084508084373056203, −9.131443655706702216676646600498, −8.244831465099372749327358133903, −7.38675805265098210708587232252, −6.41440556897093580869811851316, −4.89580709677700674330960556883, −3.95964851266685179088986808677, −2.61290832881892085720207796952, 0,
2.61290832881892085720207796952, 3.95964851266685179088986808677, 4.89580709677700674330960556883, 6.41440556897093580869811851316, 7.38675805265098210708587232252, 8.244831465099372749327358133903, 9.131443655706702216676646600498, 10.45530561787084508084373056203, 11.13154017588271625556961179667
