| L(s) = 1 | + 2-s + 4-s + 4.32·5-s − 4.89·7-s + 8-s + 4.32·10-s − 2.42·13-s − 4.89·14-s + 16-s − 4.15·19-s + 4.32·20-s − 7.32·23-s + 13.6·25-s − 2.42·26-s − 4.89·28-s − 0.284·29-s − 6.48·31-s + 32-s − 21.1·35-s − 7.19·37-s − 4.15·38-s + 4.32·40-s − 8.48·41-s + 2.48·43-s − 7.32·46-s − 10.7·47-s + 16.9·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.93·5-s − 1.85·7-s + 0.353·8-s + 1.36·10-s − 0.672·13-s − 1.30·14-s + 0.250·16-s − 0.953·19-s + 0.966·20-s − 1.52·23-s + 2.73·25-s − 0.475·26-s − 0.925·28-s − 0.0528·29-s − 1.16·31-s + 0.176·32-s − 3.57·35-s − 1.18·37-s − 0.674·38-s + 0.683·40-s − 1.32·41-s + 0.378·43-s − 1.07·46-s − 1.57·47-s + 2.42·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 4.32T + 5T^{2} \) |
| 7 | \( 1 + 4.89T + 7T^{2} \) |
| 13 | \( 1 + 2.42T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 4.15T + 19T^{2} \) |
| 23 | \( 1 + 7.32T + 23T^{2} \) |
| 29 | \( 1 + 0.284T + 29T^{2} \) |
| 31 | \( 1 + 6.48T + 31T^{2} \) |
| 37 | \( 1 + 7.19T + 37T^{2} \) |
| 41 | \( 1 + 8.48T + 41T^{2} \) |
| 43 | \( 1 - 2.48T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 - 6.49T + 53T^{2} \) |
| 59 | \( 1 + 4.36T + 59T^{2} \) |
| 61 | \( 1 - 5.64T + 61T^{2} \) |
| 67 | \( 1 + 2.28T + 67T^{2} \) |
| 71 | \( 1 - 2.64T + 71T^{2} \) |
| 73 | \( 1 + 0.394T + 73T^{2} \) |
| 79 | \( 1 + 8.30T + 79T^{2} \) |
| 83 | \( 1 + 5.00T + 83T^{2} \) |
| 89 | \( 1 - 3.51T + 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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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
−7.24343726017283549481699597365, −6.63736513511282019548195829924, −6.21699985354910290271539666860, −5.64741521801517750219330940108, −5.00463548735910563641555883110, −3.89596863944097511962220174796, −3.13720309451621046107321253619, −2.33833409982382003384183812629, −1.75437532748665162796621178799, 0,
1.75437532748665162796621178799, 2.33833409982382003384183812629, 3.13720309451621046107321253619, 3.89596863944097511962220174796, 5.00463548735910563641555883110, 5.64741521801517750219330940108, 6.21699985354910290271539666860, 6.63736513511282019548195829924, 7.24343726017283549481699597365
