| L(s) = 1 | − 1.23·2-s + 2.26·3-s − 0.486·4-s − 5-s − 2.78·6-s − 7-s + 3.05·8-s + 2.11·9-s + 1.23·10-s + 0.0423·11-s − 1.10·12-s + 2.92·13-s + 1.23·14-s − 2.26·15-s − 2.78·16-s − 2.56·17-s − 2.60·18-s + 3.83·19-s + 0.486·20-s − 2.26·21-s − 0.0521·22-s − 1.59·23-s + 6.92·24-s + 25-s − 3.60·26-s − 1.99·27-s + 0.486·28-s + ⋯ |
| L(s) = 1 | − 0.869·2-s + 1.30·3-s − 0.243·4-s − 0.447·5-s − 1.13·6-s − 0.377·7-s + 1.08·8-s + 0.706·9-s + 0.389·10-s + 0.0127·11-s − 0.317·12-s + 0.812·13-s + 0.328·14-s − 0.584·15-s − 0.697·16-s − 0.621·17-s − 0.614·18-s + 0.878·19-s + 0.108·20-s − 0.493·21-s − 0.0111·22-s − 0.331·23-s + 1.41·24-s + 0.200·25-s − 0.706·26-s − 0.383·27-s + 0.0919·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 + T \) |
| good | 2 | \( 1 + 1.23T + 2T^{2} \) |
| 3 | \( 1 - 2.26T + 3T^{2} \) |
| 11 | \( 1 - 0.0423T + 11T^{2} \) |
| 13 | \( 1 - 2.92T + 13T^{2} \) |
| 17 | \( 1 + 2.56T + 17T^{2} \) |
| 19 | \( 1 - 3.83T + 19T^{2} \) |
| 23 | \( 1 + 1.59T + 23T^{2} \) |
| 29 | \( 1 + 1.44T + 29T^{2} \) |
| 31 | \( 1 + 2.18T + 31T^{2} \) |
| 37 | \( 1 + 0.287T + 37T^{2} \) |
| 41 | \( 1 + 0.141T + 41T^{2} \) |
| 43 | \( 1 + 8.41T + 43T^{2} \) |
| 47 | \( 1 - 8.99T + 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 - 9.00T + 59T^{2} \) |
| 61 | \( 1 + 6.11T + 61T^{2} \) |
| 67 | \( 1 - 5.12T + 67T^{2} \) |
| 71 | \( 1 + 9.43T + 71T^{2} \) |
| 73 | \( 1 + 4.32T + 73T^{2} \) |
| 79 | \( 1 + 7.67T + 79T^{2} \) |
| 83 | \( 1 - 1.48T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 + 1.02T + 97T^{2} \) |
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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
−7.70504580388031308635205961674, −7.23041215507859690703348386209, −6.32982203496946928318959922478, −5.34424324659273815040451247049, −4.40433310876048537933059666829, −3.72491096736127426652971938502, −3.15671664688994426793073129129, −2.13471612082050696553945183720, −1.24583084985099884517425409693, 0,
1.24583084985099884517425409693, 2.13471612082050696553945183720, 3.15671664688994426793073129129, 3.72491096736127426652971938502, 4.40433310876048537933059666829, 5.34424324659273815040451247049, 6.32982203496946928318959922478, 7.23041215507859690703348386209, 7.70504580388031308635205961674
