| L(s) = 1 | − 2·7-s − 11-s − 2·13-s − 2·17-s − 6·19-s + 4·23-s + 6·29-s − 4·31-s − 6·37-s − 10·41-s − 6·43-s − 8·47-s − 3·49-s + 4·59-s + 6·61-s − 8·67-s + 2·73-s + 2·77-s + 10·79-s − 12·83-s + 4·91-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 0.301·11-s − 0.554·13-s − 0.485·17-s − 1.37·19-s + 0.834·23-s + 1.11·29-s − 0.718·31-s − 0.986·37-s − 1.56·41-s − 0.914·43-s − 1.16·47-s − 3/7·49-s + 0.520·59-s + 0.768·61-s − 0.977·67-s + 0.234·73-s + 0.227·77-s + 1.12·79-s − 1.31·83-s + 0.419·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 158400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 158400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−13.60382118329147, −13.18519593539807, −12.90736317098656, −12.41364939879864, −11.90263170572372, −11.41340385240186, −10.85368686327099, −10.28353814674103, −10.11880929345870, −9.462776305779552, −8.931321004318653, −8.450982332092644, −8.142168925733525, −7.300989966375499, −6.830906526637930, −6.555219181000554, −6.037014901686780, −5.108463079855215, −5.009239589649395, −4.302213209016204, −3.572870920079907, −3.184274278421682, −2.494817617140091, −1.965759363241553, −1.224622175165205, 0, 0,
1.224622175165205, 1.965759363241553, 2.494817617140091, 3.184274278421682, 3.572870920079907, 4.302213209016204, 5.009239589649395, 5.108463079855215, 6.037014901686780, 6.555219181000554, 6.830906526637930, 7.300989966375499, 8.142168925733525, 8.450982332092644, 8.931321004318653, 9.462776305779552, 10.11880929345870, 10.28353814674103, 10.85368686327099, 11.41340385240186, 11.90263170572372, 12.41364939879864, 12.90736317098656, 13.18519593539807, 13.60382118329147
