| L(s) = 1 | − 5-s + 4.73·7-s − 5.73·11-s + 1.46·13-s + 2.73·17-s − 4.46·19-s − 3.46·23-s + 25-s − 3.19·29-s + 3·31-s − 4.73·35-s − 2.73·37-s + 7.19·41-s − 0.196·43-s − 8.73·47-s + 15.3·49-s − 6.73·53-s + 5.73·55-s − 8.26·59-s + 4·61-s − 1.46·65-s − 3.46·67-s − 3.73·71-s − 7.66·73-s − 27.1·77-s − 15.4·79-s + 2.19·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.78·7-s − 1.72·11-s + 0.406·13-s + 0.662·17-s − 1.02·19-s − 0.722·23-s + 0.200·25-s − 0.593·29-s + 0.538·31-s − 0.799·35-s − 0.449·37-s + 1.12·41-s − 0.0299·43-s − 1.27·47-s + 2.19·49-s − 0.924·53-s + 0.772·55-s − 1.07·59-s + 0.512·61-s − 0.181·65-s − 0.423·67-s − 0.442·71-s − 0.896·73-s − 3.09·77-s − 1.73·79-s + 0.241·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| good | 7 | \( 1 - 4.73T + 7T^{2} \) |
| 11 | \( 1 + 5.73T + 11T^{2} \) |
| 13 | \( 1 - 1.46T + 13T^{2} \) |
| 17 | \( 1 - 2.73T + 17T^{2} \) |
| 19 | \( 1 + 4.46T + 19T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 + 3.19T + 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 + 2.73T + 37T^{2} \) |
| 41 | \( 1 - 7.19T + 41T^{2} \) |
| 43 | \( 1 + 0.196T + 43T^{2} \) |
| 47 | \( 1 + 8.73T + 47T^{2} \) |
| 53 | \( 1 + 6.73T + 53T^{2} \) |
| 59 | \( 1 + 8.26T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 + 3.46T + 67T^{2} \) |
| 71 | \( 1 + 3.73T + 71T^{2} \) |
| 73 | \( 1 + 7.66T + 73T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 - 2.19T + 83T^{2} \) |
| 89 | \( 1 + 5.19T + 89T^{2} \) |
| 97 | \( 1 + 9.66T + 97T^{2} \) |
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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
−7.77139274317460706441952665695, −7.28836396498341815640593765038, −6.07377004795538597572966848227, −5.46236005833839902908917215016, −4.72349880341018010825173570683, −4.24694990166014620227632811423, −3.13305081349511119960135942463, −2.22315668380745261981357809017, −1.40691898209322406080033843097, 0,
1.40691898209322406080033843097, 2.22315668380745261981357809017, 3.13305081349511119960135942463, 4.24694990166014620227632811423, 4.72349880341018010825173570683, 5.46236005833839902908917215016, 6.07377004795538597572966848227, 7.28836396498341815640593765038, 7.77139274317460706441952665695
