| L(s) = 1 | − 20·7-s + 56·11-s + 86·13-s − 106·17-s + 4·19-s + 136·23-s + 206·29-s − 152·31-s − 282·37-s + 246·41-s − 412·43-s + 40·47-s + 57·49-s − 126·53-s − 56·59-s − 2·61-s + 388·67-s + 672·71-s − 1.17e3·73-s − 1.12e3·77-s + 408·79-s + 668·83-s − 66·89-s − 1.72e3·91-s + 926·97-s + 198·101-s + 1.53e3·103-s + ⋯ |
| L(s) = 1 | − 1.07·7-s + 1.53·11-s + 1.83·13-s − 1.51·17-s + 0.0482·19-s + 1.23·23-s + 1.31·29-s − 0.880·31-s − 1.25·37-s + 0.937·41-s − 1.46·43-s + 0.124·47-s + 0.166·49-s − 0.326·53-s − 0.123·59-s − 0.00419·61-s + 0.707·67-s + 1.12·71-s − 1.87·73-s − 1.65·77-s + 0.581·79-s + 0.883·83-s − 0.0786·89-s − 1.98·91-s + 0.969·97-s + 0.195·101-s + 1.46·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.179718883\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.179718883\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 + 20 T + p^{3} T^{2} \) |
| 11 | \( 1 - 56 T + p^{3} T^{2} \) |
| 13 | \( 1 - 86 T + p^{3} T^{2} \) |
| 17 | \( 1 + 106 T + p^{3} T^{2} \) |
| 19 | \( 1 - 4 T + p^{3} T^{2} \) |
| 23 | \( 1 - 136 T + p^{3} T^{2} \) |
| 29 | \( 1 - 206 T + p^{3} T^{2} \) |
| 31 | \( 1 + 152 T + p^{3} T^{2} \) |
| 37 | \( 1 + 282 T + p^{3} T^{2} \) |
| 41 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 43 | \( 1 + 412 T + p^{3} T^{2} \) |
| 47 | \( 1 - 40 T + p^{3} T^{2} \) |
| 53 | \( 1 + 126 T + p^{3} T^{2} \) |
| 59 | \( 1 + 56 T + p^{3} T^{2} \) |
| 61 | \( 1 + 2 T + p^{3} T^{2} \) |
| 67 | \( 1 - 388 T + p^{3} T^{2} \) |
| 71 | \( 1 - 672 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1170 T + p^{3} T^{2} \) |
| 79 | \( 1 - 408 T + p^{3} T^{2} \) |
| 83 | \( 1 - 668 T + p^{3} T^{2} \) |
| 89 | \( 1 + 66 T + p^{3} T^{2} \) |
| 97 | \( 1 - 926 T + p^{3} T^{2} \) |
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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
−8.925312413838315296153567844285, −8.455320185200340270240312583880, −6.95383968431445259905055162589, −6.59496316899241756784596515146, −5.98217654583354121269046679930, −4.69809004211638301673652274808, −3.75589351529804126117477206645, −3.18847020563996099663551912861, −1.74854948086973600117930304728, −0.71893256481158632138902772625,
0.71893256481158632138902772625, 1.74854948086973600117930304728, 3.18847020563996099663551912861, 3.75589351529804126117477206645, 4.69809004211638301673652274808, 5.98217654583354121269046679930, 6.59496316899241756784596515146, 6.95383968431445259905055162589, 8.455320185200340270240312583880, 8.925312413838315296153567844285
