| L(s) = 1 | − 0.267·5-s − 3.46·7-s − 2·11-s + 4.46·13-s − 5.73·17-s − 0.535·19-s − 8.92·23-s − 4.92·25-s − 7.73·29-s + 2.92·31-s + 0.928·35-s + 6.46·37-s + 6.92·41-s − 11.4·43-s + 6.92·47-s + 4.99·49-s − 2.92·53-s + 0.535·55-s − 8·59-s + 3.53·61-s − 1.19·65-s − 7.46·67-s + 2·71-s + 73-s + 6.92·77-s + 7.46·79-s − 10.9·83-s + ⋯ |
| L(s) = 1 | − 0.119·5-s − 1.30·7-s − 0.603·11-s + 1.23·13-s − 1.39·17-s − 0.122·19-s − 1.86·23-s − 0.985·25-s − 1.43·29-s + 0.525·31-s + 0.156·35-s + 1.06·37-s + 1.08·41-s − 1.74·43-s + 1.01·47-s + 0.714·49-s − 0.402·53-s + 0.0722·55-s − 1.04·59-s + 0.452·61-s − 0.148·65-s − 0.911·67-s + 0.237·71-s + 0.117·73-s + 0.789·77-s + 0.839·79-s − 1.19·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 0.267T + 5T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 4.46T + 13T^{2} \) |
| 17 | \( 1 + 5.73T + 17T^{2} \) |
| 19 | \( 1 + 0.535T + 19T^{2} \) |
| 23 | \( 1 + 8.92T + 23T^{2} \) |
| 29 | \( 1 + 7.73T + 29T^{2} \) |
| 31 | \( 1 - 2.92T + 31T^{2} \) |
| 37 | \( 1 - 6.46T + 37T^{2} \) |
| 41 | \( 1 - 6.92T + 41T^{2} \) |
| 43 | \( 1 + 11.4T + 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 + 2.92T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 - 3.53T + 61T^{2} \) |
| 67 | \( 1 + 7.46T + 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 - T + 73T^{2} \) |
| 79 | \( 1 - 7.46T + 79T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 + 5.19T + 89T^{2} \) |
| 97 | \( 1 - 15.8T + 97T^{2} \) |
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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
−10.06293077481302333226497401023, −9.340877390722703218212824294262, −8.409915020587078426871134001697, −7.50374129940488760068670543259, −6.29496994419740868708539797369, −5.92492540220685205058599080107, −4.31037293641946565348253159682, −3.47870866874595654812355866409, −2.16050447840299758620539758585, 0,
2.16050447840299758620539758585, 3.47870866874595654812355866409, 4.31037293641946565348253159682, 5.92492540220685205058599080107, 6.29496994419740868708539797369, 7.50374129940488760068670543259, 8.409915020587078426871134001697, 9.340877390722703218212824294262, 10.06293077481302333226497401023
