| L(s) = 1 | − 5-s − 7-s + 11-s − 13-s + 5·17-s − 3·23-s + 25-s + 2·29-s − 6·31-s + 35-s + 37-s + 7·41-s + 2·43-s + 6·47-s − 6·49-s + 5·53-s − 55-s + 4·59-s − 7·61-s + 65-s + 12·67-s + 71-s + 10·73-s − 77-s + 11·79-s + 12·83-s − 5·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s + 0.301·11-s − 0.277·13-s + 1.21·17-s − 0.625·23-s + 1/5·25-s + 0.371·29-s − 1.07·31-s + 0.169·35-s + 0.164·37-s + 1.09·41-s + 0.304·43-s + 0.875·47-s − 6/7·49-s + 0.686·53-s − 0.134·55-s + 0.520·59-s − 0.896·61-s + 0.124·65-s + 1.46·67-s + 0.118·71-s + 1.17·73-s − 0.113·77-s + 1.23·79-s + 1.31·83-s − 0.542·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.501844655\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.501844655\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 - 7 T + p T^{2} \) | 1.41.ah |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 5 T + p T^{2} \) | 1.53.af |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - T + p T^{2} \) | 1.71.ab |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 13 T + p T^{2} \) | 1.89.an |
| 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
−9.098186629469526853151658183392, −8.062276134701930237978779824513, −7.57196959268051820934545497771, −6.69412594520033315755042481157, −5.87260904722898126690054128166, −5.05591228492984513329523909335, −4.00048372273859856959658740977, −3.34767981023045646795364093942, −2.20644842512625434144019888859, −0.790324837918246793898471311545,
0.790324837918246793898471311545, 2.20644842512625434144019888859, 3.34767981023045646795364093942, 4.00048372273859856959658740977, 5.05591228492984513329523909335, 5.87260904722898126690054128166, 6.69412594520033315755042481157, 7.57196959268051820934545497771, 8.062276134701930237978779824513, 9.098186629469526853151658183392
