| L(s) = 1 | − 2·7-s − 6·11-s + 2·13-s + 23-s − 6·29-s − 8·31-s − 10·41-s − 12·43-s + 8·47-s − 3·49-s + 2·53-s − 12·59-s + 4·61-s − 12·67-s + 10·73-s + 12·77-s + 6·79-s − 14·83-s − 4·91-s + 6·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 1.80·11-s + 0.554·13-s + 0.208·23-s − 1.11·29-s − 1.43·31-s − 1.56·41-s − 1.82·43-s + 1.16·47-s − 3/7·49-s + 0.274·53-s − 1.56·59-s + 0.512·61-s − 1.46·67-s + 1.17·73-s + 1.36·77-s + 0.675·79-s − 1.53·83-s − 0.419·91-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−14.55442969450337, −13.68574391469892, −13.35575456367357, −13.13516779657390, −12.54645362535843, −12.10672639197926, −11.34529862735815, −10.95563206617841, −10.41039662821082, −10.09065421603856, −9.431983867937576, −8.964741728213457, −8.361153256779932, −7.875932396071626, −7.316623324449891, −6.873835807199007, −6.165988424912287, −5.649237416969200, −5.168762210775963, −4.675245466953169, −3.656547877538053, −3.425903419994192, −2.721705045456114, −2.058346829750934, −1.353227750518666, 0, 0,
1.353227750518666, 2.058346829750934, 2.721705045456114, 3.425903419994192, 3.656547877538053, 4.675245466953169, 5.168762210775963, 5.649237416969200, 6.165988424912287, 6.873835807199007, 7.316623324449891, 7.875932396071626, 8.361153256779932, 8.964741728213457, 9.431983867937576, 10.09065421603856, 10.41039662821082, 10.95563206617841, 11.34529862735815, 12.10672639197926, 12.54645362535843, 13.13516779657390, 13.35575456367357, 13.68574391469892, 14.55442969450337
