| L(s) = 1 | + 3·5-s − 5·7-s − 3·11-s − 4·13-s − 6·17-s − 19-s + 6·23-s + 4·25-s − 6·29-s + 7·31-s − 15·35-s − 4·37-s + 4·43-s + 6·47-s + 18·49-s − 9·53-s − 9·55-s + 14·61-s − 12·65-s − 8·67-s − 6·71-s + 5·73-s + 15·77-s − 8·79-s + 9·83-s − 18·85-s + 6·89-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 1.88·7-s − 0.904·11-s − 1.10·13-s − 1.45·17-s − 0.229·19-s + 1.25·23-s + 4/5·25-s − 1.11·29-s + 1.25·31-s − 2.53·35-s − 0.657·37-s + 0.609·43-s + 0.875·47-s + 18/7·49-s − 1.23·53-s − 1.21·55-s + 1.79·61-s − 1.48·65-s − 0.977·67-s − 0.712·71-s + 0.585·73-s + 1.70·77-s − 0.900·79-s + 0.987·83-s − 1.95·85-s + 0.635·89-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
| \(L(1)\) |
\(\approx\) |
\(1.183233091\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.183233091\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 + 5 T + p T^{2} \) | 1.7.f |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 5 T + p T^{2} \) | 1.73.af |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 17 T + p T^{2} \) | 1.97.ar |
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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
−7.61900254483713665351219929368, −6.91219671715355334982044889349, −6.48325789933692785024104199911, −5.80617447813501160063097869884, −5.17068200556449795431623141952, −4.36528570555324090366372511742, −3.22783512191396845451797733696, −2.59731756362079253207088675638, −2.08059409670364561370336341002, −0.49521450682313681861383114765,
0.49521450682313681861383114765, 2.08059409670364561370336341002, 2.59731756362079253207088675638, 3.22783512191396845451797733696, 4.36528570555324090366372511742, 5.17068200556449795431623141952, 5.80617447813501160063097869884, 6.48325789933692785024104199911, 6.91219671715355334982044889349, 7.61900254483713665351219929368
