| L(s) = 1 | − 3-s − 2·5-s + 7-s + 9-s + 11-s − 2·13-s + 2·15-s + 2·17-s − 4·19-s − 21-s − 25-s − 27-s + 6·29-s + 8·31-s − 33-s − 2·35-s − 10·37-s + 2·39-s − 6·41-s + 12·43-s − 2·45-s + 8·47-s + 49-s − 2·51-s − 10·53-s − 2·55-s + 4·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.554·13-s + 0.516·15-s + 0.485·17-s − 0.917·19-s − 0.218·21-s − 1/5·25-s − 0.192·27-s + 1.11·29-s + 1.43·31-s − 0.174·33-s − 0.338·35-s − 1.64·37-s + 0.320·39-s − 0.937·41-s + 1.82·43-s − 0.298·45-s + 1.16·47-s + 1/7·49-s − 0.280·51-s − 1.37·53-s − 0.269·55-s + 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−8.068663878888556090842327303545, −7.47094770535678434143849360916, −6.67205328908414648275978054665, −5.98822662661189458881738940877, −4.96063693031033258562912631739, −4.43364113772305065865530659782, −3.61363294811631251804216626231, −2.51363413087862243514806606391, −1.25268535851082751439809001964, 0,
1.25268535851082751439809001964, 2.51363413087862243514806606391, 3.61363294811631251804216626231, 4.43364113772305065865530659782, 4.96063693031033258562912631739, 5.98822662661189458881738940877, 6.67205328908414648275978054665, 7.47094770535678434143849360916, 8.068663878888556090842327303545
