| L(s) = 1 | + 3-s + 5-s − 3·7-s + 9-s − 2·13-s + 15-s − 4·17-s + 7·19-s − 3·21-s − 2·23-s + 25-s + 27-s − 6·29-s + 31-s − 3·35-s − 11·37-s − 2·39-s − 8·41-s + 4·43-s + 45-s − 4·47-s + 2·49-s − 4·51-s + 2·53-s + 7·57-s + 2·59-s − 5·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.13·7-s + 1/3·9-s − 0.554·13-s + 0.258·15-s − 0.970·17-s + 1.60·19-s − 0.654·21-s − 0.417·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s + 0.179·31-s − 0.507·35-s − 1.80·37-s − 0.320·39-s − 1.24·41-s + 0.609·43-s + 0.149·45-s − 0.583·47-s + 2/7·49-s − 0.560·51-s + 0.274·53-s + 0.927·57-s + 0.260·59-s − 0.640·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 + 11 T + p T^{2} \) | 1.37.l |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 + T + p T^{2} \) | 1.67.b |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 9 T + p T^{2} \) | 1.97.aj |
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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
−13.66462322862619, −13.50512709074451, −13.01462670780389, −12.36569480315812, −12.09355611132992, −11.46029416128054, −10.83480521190094, −10.27242277486376, −9.825510055192443, −9.472546386296579, −9.044437155347442, −8.565401651266849, −7.852795517720748, −7.307658930993145, −6.933340411094517, −6.381199368341521, −5.859182482019860, −5.128612698011389, −4.825439879168166, −3.886958861283846, −3.345425333982713, −3.103523943746029, −2.109814896963931, −1.917135978791390, −0.8174807185888538, 0,
0.8174807185888538, 1.917135978791390, 2.109814896963931, 3.103523943746029, 3.345425333982713, 3.886958861283846, 4.825439879168166, 5.128612698011389, 5.859182482019860, 6.381199368341521, 6.933340411094517, 7.307658930993145, 7.852795517720748, 8.565401651266849, 9.044437155347442, 9.472546386296579, 9.825510055192443, 10.27242277486376, 10.83480521190094, 11.46029416128054, 12.09355611132992, 12.36569480315812, 13.01462670780389, 13.50512709074451, 13.66462322862619
