| L(s) = 1 | + 3-s + 5-s + 4·7-s + 9-s − 2·13-s + 15-s + 2·17-s − 4·19-s + 4·21-s + 25-s + 27-s − 6·29-s − 8·31-s + 4·35-s + 2·37-s − 2·39-s + 2·41-s − 4·43-s + 45-s + 9·49-s + 2·51-s + 6·53-s − 4·57-s + 4·59-s − 2·61-s + 4·63-s − 2·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s − 0.554·13-s + 0.258·15-s + 0.485·17-s − 0.917·19-s + 0.872·21-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 1.43·31-s + 0.676·35-s + 0.328·37-s − 0.320·39-s + 0.312·41-s − 0.609·43-s + 0.149·45-s + 9/7·49-s + 0.280·51-s + 0.824·53-s − 0.529·57-s + 0.520·59-s − 0.256·61-s + 0.503·63-s − 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 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.g |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−14.58212262632948, −14.33466540115781, −13.60760294096781, −13.14342593441544, −12.67729587008563, −12.08271687691793, −11.45962484152144, −11.09345710100870, −10.43900138365889, −10.08767074248650, −9.255359664463345, −9.022042881968673, −8.276588900030443, −7.975783891116212, −7.274964234918759, −7.001122519524739, −6.005944152515399, −5.495837291228734, −5.030064684088952, −4.299301855457275, −3.908391599559557, −3.000290399526769, −2.297134523727414, −1.789425472834013, −1.252283606428136, 0,
1.252283606428136, 1.789425472834013, 2.297134523727414, 3.000290399526769, 3.908391599559557, 4.299301855457275, 5.030064684088952, 5.495837291228734, 6.005944152515399, 7.001122519524739, 7.274964234918759, 7.975783891116212, 8.276588900030443, 9.022042881968673, 9.255359664463345, 10.08767074248650, 10.43900138365889, 11.09345710100870, 11.45962484152144, 12.08271687691793, 12.67729587008563, 13.14342593441544, 13.60760294096781, 14.33466540115781, 14.58212262632948
