| L(s) = 1 | + 1.48·2-s + 0.193·4-s + 1.48·5-s − 4.15·7-s − 2.67·8-s + 2.19·10-s − 2.80·11-s − 4.35·13-s − 6.15·14-s − 4.35·16-s − 4.96·17-s + 7.35·19-s + 0.287·20-s − 4.15·22-s − 0.518·23-s − 2.80·25-s − 6.44·26-s − 0.806·28-s − 3.76·29-s + 5.96·31-s − 1.09·32-s − 7.35·34-s − 6.15·35-s + 5.11·37-s + 10.8·38-s − 3.96·40-s + 2.54·41-s + ⋯ |
| L(s) = 1 | + 1.04·2-s + 0.0969·4-s + 0.662·5-s − 1.57·7-s − 0.945·8-s + 0.693·10-s − 0.846·11-s − 1.20·13-s − 1.64·14-s − 1.08·16-s − 1.20·17-s + 1.68·19-s + 0.0642·20-s − 0.886·22-s − 0.108·23-s − 0.561·25-s − 1.26·26-s − 0.152·28-s − 0.699·29-s + 1.07·31-s − 0.193·32-s − 1.26·34-s − 1.04·35-s + 0.841·37-s + 1.76·38-s − 0.626·40-s + 0.397·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 - 1.48T + 2T^{2} \) |
| 5 | \( 1 - 1.48T + 5T^{2} \) |
| 7 | \( 1 + 4.15T + 7T^{2} \) |
| 11 | \( 1 + 2.80T + 11T^{2} \) |
| 13 | \( 1 + 4.35T + 13T^{2} \) |
| 17 | \( 1 + 4.96T + 17T^{2} \) |
| 19 | \( 1 - 7.35T + 19T^{2} \) |
| 23 | \( 1 + 0.518T + 23T^{2} \) |
| 29 | \( 1 + 3.76T + 29T^{2} \) |
| 31 | \( 1 - 5.96T + 31T^{2} \) |
| 37 | \( 1 - 5.11T + 37T^{2} \) |
| 41 | \( 1 - 2.54T + 41T^{2} \) |
| 43 | \( 1 - 3.15T + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 - 6.15T + 53T^{2} \) |
| 59 | \( 1 + 2.90T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 + 7.70T + 67T^{2} \) |
| 71 | \( 1 - 7.66T + 71T^{2} \) |
| 73 | \( 1 + 11.1T + 73T^{2} \) |
| 79 | \( 1 - 7.50T + 79T^{2} \) |
| 83 | \( 1 + 3.71T + 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
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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
−9.553393734079097293459654174306, −9.259330220299460973781916139261, −7.77607401379022943280937759689, −6.77792024673962448071928612009, −5.99618369891304856922267907430, −5.28670631915244665824282212121, −4.34856285913009715228938314768, −3.13013771850813316309912820585, −2.51782899878293952313674986829, 0,
2.51782899878293952313674986829, 3.13013771850813316309912820585, 4.34856285913009715228938314768, 5.28670631915244665824282212121, 5.99618369891304856922267907430, 6.77792024673962448071928612009, 7.77607401379022943280937759689, 9.259330220299460973781916139261, 9.553393734079097293459654174306
