| L(s) = 1 | + 2·7-s + 5·11-s − 4·13-s + 2·17-s + 19-s − 8·23-s − 5·25-s + 29-s + 3·31-s − 3·41-s + 10·43-s + 9·47-s − 3·49-s + 13·53-s − 4·59-s + 5·61-s + 67-s + 6·71-s + 7·73-s + 10·77-s + 11·79-s − 15·83-s − 3·89-s − 8·91-s + 8·97-s + 8·101-s + 8·103-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 1.50·11-s − 1.10·13-s + 0.485·17-s + 0.229·19-s − 1.66·23-s − 25-s + 0.185·29-s + 0.538·31-s − 0.468·41-s + 1.52·43-s + 1.31·47-s − 3/7·49-s + 1.78·53-s − 0.520·59-s + 0.640·61-s + 0.122·67-s + 0.712·71-s + 0.819·73-s + 1.13·77-s + 1.23·79-s − 1.64·83-s − 0.317·89-s − 0.838·91-s + 0.812·97-s + 0.796·101-s + 0.788·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.354691456\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.354691456\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 19 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 - 13 T + p T^{2} \) | 1.53.an |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 - T + p T^{2} \) | 1.67.ab |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 + 15 T + p T^{2} \) | 1.83.p |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−7.72447849884197959664571698094, −7.27839377802426734654573428776, −6.37664365578384530277969522357, −5.78385797725944969501811721071, −4.99962843208792518629858962956, −4.15847532233669585019542958490, −3.75465881913702910471207817845, −2.48728183627205780152279628230, −1.80630547274792375850547470308, −0.77404720552156091863312297047,
0.77404720552156091863312297047, 1.80630547274792375850547470308, 2.48728183627205780152279628230, 3.75465881913702910471207817845, 4.15847532233669585019542958490, 4.99962843208792518629858962956, 5.78385797725944969501811721071, 6.37664365578384530277969522357, 7.27839377802426734654573428776, 7.72447849884197959664571698094
