| L(s) = 1 | + 3·5-s + 7-s + 3·11-s + 4·13-s + 2·19-s − 6·23-s + 4·25-s + 6·29-s − 5·31-s + 3·35-s − 2·37-s + 6·41-s − 10·43-s + 6·47-s − 6·49-s + 9·53-s + 9·55-s − 12·59-s − 8·61-s + 12·65-s + 14·67-s − 7·73-s + 3·77-s − 8·79-s + 3·83-s + 18·89-s + 4·91-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 0.377·7-s + 0.904·11-s + 1.10·13-s + 0.458·19-s − 1.25·23-s + 4/5·25-s + 1.11·29-s − 0.898·31-s + 0.507·35-s − 0.328·37-s + 0.937·41-s − 1.52·43-s + 0.875·47-s − 6/7·49-s + 1.23·53-s + 1.21·55-s − 1.56·59-s − 1.02·61-s + 1.48·65-s + 1.71·67-s − 0.819·73-s + 0.341·77-s − 0.900·79-s + 0.329·83-s + 1.90·89-s + 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.577750507\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.577750507\) |
| \(L(\frac{3}{2})\) |
|
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.ad |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−9.300538826904107531692188360431, −8.699511810288277483701861071116, −7.82087500680797002813709866504, −6.68641118928175950907657181471, −6.10265864740654543835243104495, −5.43217621063437274714591238543, −4.34994968190190118318545746782, −3.35914372705473359723249826168, −2.05899421145709367031472603742, −1.26980488075185704480426069120,
1.26980488075185704480426069120, 2.05899421145709367031472603742, 3.35914372705473359723249826168, 4.34994968190190118318545746782, 5.43217621063437274714591238543, 6.10265864740654543835243104495, 6.68641118928175950907657181471, 7.82087500680797002813709866504, 8.699511810288277483701861071116, 9.300538826904107531692188360431
