| L(s) = 1 | − 3-s + 2·5-s + 2·7-s + 9-s − 6·11-s + 2·13-s − 2·15-s − 2·21-s + 23-s − 25-s − 27-s − 6·29-s − 8·31-s + 6·33-s + 4·35-s − 2·39-s + 10·41-s − 12·43-s + 2·45-s + 8·47-s − 3·49-s − 2·53-s − 12·55-s − 12·59-s − 4·61-s + 2·63-s + 4·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 0.755·7-s + 1/3·9-s − 1.80·11-s + 0.554·13-s − 0.516·15-s − 0.436·21-s + 0.208·23-s − 1/5·25-s − 0.192·27-s − 1.11·29-s − 1.43·31-s + 1.04·33-s + 0.676·35-s − 0.320·39-s + 1.56·41-s − 1.82·43-s + 0.298·45-s + 1.16·47-s − 3/7·49-s − 0.274·53-s − 1.61·55-s − 1.56·59-s − 0.512·61-s + 0.251·63-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−7.72047880217154070655817213438, −7.52450739343745495455790166819, −6.30687778302223289318527264405, −5.66537004424004383201735558506, −5.23091445179449772114465442965, −4.45808410287090706931387677434, −3.30037715777891240894020217870, −2.23440726888925536869693203702, −1.51685500861613377868289358777, 0,
1.51685500861613377868289358777, 2.23440726888925536869693203702, 3.30037715777891240894020217870, 4.45808410287090706931387677434, 5.23091445179449772114465442965, 5.66537004424004383201735558506, 6.30687778302223289318527264405, 7.52450739343745495455790166819, 7.72047880217154070655817213438
