| L(s) = 1 | − 1.73·5-s − 2·7-s + 3.46·11-s − 13-s + 5.19·17-s − 2·19-s − 3.46·23-s − 2.00·25-s + 1.73·29-s − 8·31-s + 3.46·35-s − 7·37-s − 6.92·41-s − 2·43-s − 6.92·47-s − 3·49-s − 5.99·55-s − 13.8·59-s − 7·61-s + 1.73·65-s + 10·67-s + 10.3·71-s − 7·73-s − 6.92·77-s − 2·79-s + 13.8·83-s − 9·85-s + ⋯ |
| L(s) = 1 | − 0.774·5-s − 0.755·7-s + 1.04·11-s − 0.277·13-s + 1.26·17-s − 0.458·19-s − 0.722·23-s − 0.400·25-s + 0.321·29-s − 1.43·31-s + 0.585·35-s − 1.15·37-s − 1.08·41-s − 0.304·43-s − 1.01·47-s − 0.428·49-s − 0.809·55-s − 1.80·59-s − 0.896·61-s + 0.214·65-s + 1.22·67-s + 1.23·71-s − 0.819·73-s − 0.789·77-s − 0.225·79-s + 1.52·83-s − 0.976·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 1.73T + 5T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 - 5.19T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 - 1.73T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 7T + 37T^{2} \) |
| 41 | \( 1 + 6.92T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 13.8T + 59T^{2} \) |
| 61 | \( 1 + 7T + 61T^{2} \) |
| 67 | \( 1 - 10T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 + 7T + 73T^{2} \) |
| 79 | \( 1 + 2T + 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 + 5.19T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−9.341665840237041808285026809603, −8.389618765664295620574504065647, −7.63268012183850984179346547688, −6.78852207522708156626236004309, −6.03698972320889941748641237324, −4.94777782237655644122297906753, −3.78266019077985299822337629759, −3.33095289294352973300171301447, −1.69982390901808710962200993785, 0,
1.69982390901808710962200993785, 3.33095289294352973300171301447, 3.78266019077985299822337629759, 4.94777782237655644122297906753, 6.03698972320889941748641237324, 6.78852207522708156626236004309, 7.63268012183850984179346547688, 8.389618765664295620574504065647, 9.341665840237041808285026809603
