| L(s) = 1 | + 5-s − 0.732·7-s + 3.73·11-s − 5.46·13-s + 4.19·17-s − 19-s − 0.535·23-s + 25-s − 3.73·29-s − 4.46·31-s − 0.732·35-s − 10.7·37-s − 4.26·41-s + 4.19·43-s − 8.73·47-s − 6.46·49-s − 1.26·53-s + 3.73·55-s + 9.19·59-s − 1.07·61-s − 5.46·65-s + 7.46·67-s − 12.1·71-s − 4.73·73-s − 2.73·77-s + 15.4·79-s + 13.1·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.276·7-s + 1.12·11-s − 1.51·13-s + 1.01·17-s − 0.229·19-s − 0.111·23-s + 0.200·25-s − 0.693·29-s − 0.801·31-s − 0.123·35-s − 1.76·37-s − 0.666·41-s + 0.639·43-s − 1.27·47-s − 0.923·49-s − 0.174·53-s + 0.503·55-s + 1.19·59-s − 0.137·61-s − 0.677·65-s + 0.911·67-s − 1.43·71-s − 0.553·73-s − 0.311·77-s + 1.73·79-s + 1.44·83-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 | \( 1 + 0.732T + 7T^{2} \) |
| 11 | \( 1 - 3.73T + 11T^{2} \) |
| 13 | \( 1 + 5.46T + 13T^{2} \) |
| 17 | \( 1 - 4.19T + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 + 0.535T + 23T^{2} \) |
| 29 | \( 1 + 3.73T + 29T^{2} \) |
| 31 | \( 1 + 4.46T + 31T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 41 | \( 1 + 4.26T + 41T^{2} \) |
| 43 | \( 1 - 4.19T + 43T^{2} \) |
| 47 | \( 1 + 8.73T + 47T^{2} \) |
| 53 | \( 1 + 1.26T + 53T^{2} \) |
| 59 | \( 1 - 9.19T + 59T^{2} \) |
| 61 | \( 1 + 1.07T + 61T^{2} \) |
| 67 | \( 1 - 7.46T + 67T^{2} \) |
| 71 | \( 1 + 12.1T + 71T^{2} \) |
| 73 | \( 1 + 4.73T + 73T^{2} \) |
| 79 | \( 1 - 15.4T + 79T^{2} \) |
| 83 | \( 1 - 13.1T + 83T^{2} \) |
| 89 | \( 1 - 5.19T + 89T^{2} \) |
| 97 | \( 1 - 0.196T + 97T^{2} \) |
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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
−7.56684352946880687610906389896, −6.91800436575377074953010548365, −6.36594038921468195391296105839, −5.41480237799043983239576792454, −4.97465527561092328179525123490, −3.89037067612526388205818496529, −3.26698904413081970954998333747, −2.21229390545806889498018299301, −1.42445503913507547427809782019, 0,
1.42445503913507547427809782019, 2.21229390545806889498018299301, 3.26698904413081970954998333747, 3.89037067612526388205818496529, 4.97465527561092328179525123490, 5.41480237799043983239576792454, 6.36594038921468195391296105839, 6.91800436575377074953010548365, 7.56684352946880687610906389896
