| L(s) = 1 | − 3-s + 3.30·5-s − 4.71·7-s + 9-s + 1.77·11-s + 0.221·13-s − 3.30·15-s + 5.08·17-s − 1.77·19-s + 4.71·21-s + 8.96·23-s + 5.94·25-s − 27-s + 0.692·29-s + 3.55·31-s − 1.77·33-s − 15.6·35-s − 37-s − 0.221·39-s − 10.9·41-s + 4·43-s + 3.30·45-s + 9.43·47-s + 15.2·49-s − 5.08·51-s + 9.66·53-s + 5.88·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.47·5-s − 1.78·7-s + 0.333·9-s + 0.536·11-s + 0.0614·13-s − 0.854·15-s + 1.23·17-s − 0.408·19-s + 1.02·21-s + 1.87·23-s + 1.18·25-s − 0.192·27-s + 0.128·29-s + 0.638·31-s − 0.309·33-s − 2.63·35-s − 0.164·37-s − 0.0354·39-s − 1.71·41-s + 0.609·43-s + 0.493·45-s + 1.37·47-s + 2.18·49-s − 0.712·51-s + 1.32·53-s + 0.793·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.493710251\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.493710251\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| good | 5 | \( 1 - 3.30T + 5T^{2} \) |
| 7 | \( 1 + 4.71T + 7T^{2} \) |
| 11 | \( 1 - 1.77T + 11T^{2} \) |
| 13 | \( 1 - 0.221T + 13T^{2} \) |
| 17 | \( 1 - 5.08T + 17T^{2} \) |
| 19 | \( 1 + 1.77T + 19T^{2} \) |
| 23 | \( 1 - 8.96T + 23T^{2} \) |
| 29 | \( 1 - 0.692T + 29T^{2} \) |
| 31 | \( 1 - 3.55T + 31T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 9.43T + 47T^{2} \) |
| 53 | \( 1 - 9.66T + 53T^{2} \) |
| 59 | \( 1 - 4.86T + 59T^{2} \) |
| 61 | \( 1 + 6.99T + 61T^{2} \) |
| 67 | \( 1 - 4.11T + 67T^{2} \) |
| 71 | \( 1 + 5.88T + 71T^{2} \) |
| 73 | \( 1 + 5.33T + 73T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 - 7.66T + 83T^{2} \) |
| 89 | \( 1 + 2.91T + 89T^{2} \) |
| 97 | \( 1 - 12.6T + 97T^{2} \) |
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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
−10.15113060426758247657102901497, −9.419804975229876443876867198062, −8.849816534930414103255717804877, −7.17619829787741123930119366420, −6.51948345932151384199194195242, −5.90825877327875724195387168975, −5.12541978700605392769461320102, −3.61982610763441315336714831993, −2.62401620377348150839938477299, −1.05742152152223578906142911265,
1.05742152152223578906142911265, 2.62401620377348150839938477299, 3.61982610763441315336714831993, 5.12541978700605392769461320102, 5.90825877327875724195387168975, 6.51948345932151384199194195242, 7.17619829787741123930119366420, 8.849816534930414103255717804877, 9.419804975229876443876867198062, 10.15113060426758247657102901497
