| L(s) = 1 | − 5-s − 7-s − 2·11-s − 2·13-s − 2·17-s + 8·19-s − 2·23-s + 25-s + 8·29-s + 8·31-s + 35-s − 6·37-s − 6·41-s − 4·43-s + 4·47-s + 49-s + 2·55-s − 4·59-s − 2·61-s + 2·65-s + 4·67-s + 6·71-s − 6·73-s + 2·77-s − 16·83-s + 2·85-s − 6·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s − 0.603·11-s − 0.554·13-s − 0.485·17-s + 1.83·19-s − 0.417·23-s + 1/5·25-s + 1.48·29-s + 1.43·31-s + 0.169·35-s − 0.986·37-s − 0.937·41-s − 0.609·43-s + 0.583·47-s + 1/7·49-s + 0.269·55-s − 0.520·59-s − 0.256·61-s + 0.248·65-s + 0.488·67-s + 0.712·71-s − 0.702·73-s + 0.227·77-s − 1.75·83-s + 0.216·85-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| good | 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 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.q |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−7.977197141342291494066193235449, −7.07813850315708352882522051302, −6.64524736595163870360159667548, −5.57163436419997422324200354280, −4.96832053464467391533000993474, −4.18131549851569523757033959988, −3.14840468921601203786522654566, −2.63697813714156890613330231745, −1.25300802784258270262242107305, 0,
1.25300802784258270262242107305, 2.63697813714156890613330231745, 3.14840468921601203786522654566, 4.18131549851569523757033959988, 4.96832053464467391533000993474, 5.57163436419997422324200354280, 6.64524736595163870360159667548, 7.07813850315708352882522051302, 7.977197141342291494066193235449
