| L(s) = 1 | + 3-s − 5-s + 4·7-s + 9-s + 6·13-s − 15-s − 6·17-s − 4·19-s + 4·21-s + 25-s + 27-s + 2·29-s + 8·31-s − 4·35-s + 10·37-s + 6·39-s − 6·41-s − 43-s − 45-s + 8·47-s + 9·49-s − 6·51-s − 6·53-s − 4·57-s + 10·61-s + 4·63-s − 6·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s + 1.66·13-s − 0.258·15-s − 1.45·17-s − 0.917·19-s + 0.872·21-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 1.43·31-s − 0.676·35-s + 1.64·37-s + 0.960·39-s − 0.937·41-s − 0.152·43-s − 0.149·45-s + 1.16·47-s + 9/7·49-s − 0.840·51-s − 0.824·53-s − 0.529·57-s + 1.28·61-s + 0.503·63-s − 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.967628104\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.967628104\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 43 | \( 1 + T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−8.246288014474196826100400289061, −7.83855523876803379069562185094, −6.72594004494620311418103246230, −6.24830337097746224100308819384, −5.10343711738289911010277999328, −4.32482291540337688561860237887, −3.97380219928213492559514621923, −2.74826149460038832905880342401, −1.90882766851294936397009934938, −0.967493005491661078618550928622,
0.967493005491661078618550928622, 1.90882766851294936397009934938, 2.74826149460038832905880342401, 3.97380219928213492559514621923, 4.32482291540337688561860237887, 5.10343711738289911010277999328, 6.24830337097746224100308819384, 6.72594004494620311418103246230, 7.83855523876803379069562185094, 8.246288014474196826100400289061
