| L(s) = 1 | + 26.8·3-s − 66.6·5-s + 24.5·7-s + 476.·9-s − 60.9·11-s − 585.·13-s − 1.78e3·15-s + 895.·17-s − 217.·19-s + 658.·21-s − 666.·23-s + 1.31e3·25-s + 6.26e3·27-s − 1.74e3·29-s + 8.69e3·31-s − 1.63e3·33-s − 1.63e3·35-s + 1.03e4·37-s − 1.56e4·39-s − 6.90e3·41-s − 1.36e4·43-s − 3.17e4·45-s − 2.61e4·47-s − 1.62e4·49-s + 2.40e4·51-s + 6.63e3·53-s + 4.06e3·55-s + ⋯ |
| L(s) = 1 | + 1.72·3-s − 1.19·5-s + 0.189·7-s + 1.96·9-s − 0.151·11-s − 0.960·13-s − 2.05·15-s + 0.751·17-s − 0.138·19-s + 0.325·21-s − 0.262·23-s + 0.421·25-s + 1.65·27-s − 0.385·29-s + 1.62·31-s − 0.261·33-s − 0.225·35-s + 1.24·37-s − 1.65·39-s − 0.641·41-s − 1.12·43-s − 2.33·45-s − 1.72·47-s − 0.964·49-s + 1.29·51-s + 0.324·53-s + 0.181·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 257 | \( 1 - 6.60e4T \) |
| good | 3 | \( 1 - 26.8T + 243T^{2} \) |
| 5 | \( 1 + 66.6T + 3.12e3T^{2} \) |
| 7 | \( 1 - 24.5T + 1.68e4T^{2} \) |
| 11 | \( 1 + 60.9T + 1.61e5T^{2} \) |
| 13 | \( 1 + 585.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 895.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 217.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 666.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.74e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.69e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.03e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 6.90e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.36e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.61e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 6.63e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.22e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.52e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.87e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 7.86e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.29e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.29e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.29e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.61e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.46e5T + 8.58e9T^{2} \) |
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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
−8.490872819298287706179434215803, −8.006091996832301914616661287039, −7.57298087095277404772376384912, −6.59490604822622628256611585045, −4.95622894855243237217346805754, −4.16361095238326364650883266275, −3.30913015213746306054294323247, −2.61139381196306066154451674915, −1.44633448905630136967657155261, 0,
1.44633448905630136967657155261, 2.61139381196306066154451674915, 3.30913015213746306054294323247, 4.16361095238326364650883266275, 4.95622894855243237217346805754, 6.59490604822622628256611585045, 7.57298087095277404772376384912, 8.006091996832301914616661287039, 8.490872819298287706179434215803
