| L(s) = 1 | + 2-s + 4-s − 5-s + 2·7-s + 8-s − 10-s + 3·11-s + 13-s + 2·14-s + 16-s − 3·19-s − 20-s + 3·22-s + 23-s + 25-s + 26-s + 2·28-s + 29-s + 31-s + 32-s − 2·35-s + 5·37-s − 3·38-s − 40-s + 2·41-s − 9·43-s + 3·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.755·7-s + 0.353·8-s − 0.316·10-s + 0.904·11-s + 0.277·13-s + 0.534·14-s + 1/4·16-s − 0.688·19-s − 0.223·20-s + 0.639·22-s + 0.208·23-s + 1/5·25-s + 0.196·26-s + 0.377·28-s + 0.185·29-s + 0.179·31-s + 0.176·32-s − 0.338·35-s + 0.821·37-s − 0.486·38-s − 0.158·40-s + 0.312·41-s − 1.37·43-s + 0.452·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.603762891\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.603762891\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−12.57057482341631, −12.03770850437251, −11.74510499281829, −11.29597627147534, −10.94437099818472, −10.50188226314764, −9.890701026834294, −9.363016206210631, −8.885187213652587, −8.319000073975210, −7.959495111232497, −7.565696205776026, −6.851084998976116, −6.418732589610308, −6.212293892515652, −5.416376322848122, −4.896678098743030, −4.515507374888139, −4.133052528963550, −3.391272951411367, −3.192495203027821, −2.313393624456282, −1.746591466760079, −1.289151710623833, −0.4931216250129219,
0.4931216250129219, 1.289151710623833, 1.746591466760079, 2.313393624456282, 3.192495203027821, 3.391272951411367, 4.133052528963550, 4.515507374888139, 4.896678098743030, 5.416376322848122, 6.212293892515652, 6.418732589610308, 6.851084998976116, 7.565696205776026, 7.959495111232497, 8.319000073975210, 8.885187213652587, 9.363016206210631, 9.890701026834294, 10.50188226314764, 10.94437099818472, 11.29597627147534, 11.74510499281829, 12.03770850437251, 12.57057482341631
