| L(s) = 1 | + 3-s + 2·5-s + 2·7-s + 9-s + 2·15-s + 6·17-s − 6·19-s + 2·21-s + 4·23-s − 25-s + 27-s − 2·29-s + 8·31-s + 4·35-s + 2·37-s + 10·41-s − 10·43-s + 2·45-s − 3·49-s + 6·51-s + 6·53-s − 6·57-s − 4·61-s + 2·63-s + 4·67-s + 4·69-s + 4·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 0.755·7-s + 1/3·9-s + 0.516·15-s + 1.45·17-s − 1.37·19-s + 0.436·21-s + 0.834·23-s − 1/5·25-s + 0.192·27-s − 0.371·29-s + 1.43·31-s + 0.676·35-s + 0.328·37-s + 1.56·41-s − 1.52·43-s + 0.298·45-s − 3/7·49-s + 0.840·51-s + 0.824·53-s − 0.794·57-s − 0.512·61-s + 0.251·63-s + 0.488·67-s + 0.481·69-s + 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.576024720\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.576024720\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−8.101216527529362733920284979811, −7.60203709567824661521234942421, −6.62586543234615509599266466061, −6.00236403098200156751707461646, −5.19517447615078786472417917092, −4.52250570077186369522065609336, −3.59822859020526338405897249719, −2.65852423340553107777232124052, −1.92542897302579050421276274862, −1.04284445979563585882237820699,
1.04284445979563585882237820699, 1.92542897302579050421276274862, 2.65852423340553107777232124052, 3.59822859020526338405897249719, 4.52250570077186369522065609336, 5.19517447615078786472417917092, 6.00236403098200156751707461646, 6.62586543234615509599266466061, 7.60203709567824661521234942421, 8.101216527529362733920284979811
