| L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s + 2·13-s + 14-s + 16-s − 4·19-s − 20-s + 25-s − 2·26-s − 28-s − 6·29-s + 4·31-s − 32-s + 35-s − 2·37-s + 4·38-s + 40-s + 6·41-s + 8·43-s + 12·47-s + 49-s − 50-s + 2·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s − 0.353·8-s + 0.316·10-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.917·19-s − 0.223·20-s + 1/5·25-s − 0.392·26-s − 0.188·28-s − 1.11·29-s + 0.718·31-s − 0.176·32-s + 0.169·35-s − 0.328·37-s + 0.648·38-s + 0.158·40-s + 0.937·41-s + 1.21·43-s + 1.75·47-s + 1/7·49-s − 0.141·50-s + 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 182070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.227749005\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.227749005\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−12.99415061524357, −12.58282435550079, −12.29104682748032, −11.59422607198974, −11.15180683827740, −10.82967764021871, −10.34352460278080, −9.813059071371769, −9.243499155766077, −8.939112837005949, −8.327775787467330, −8.032211729603074, −7.289334264780924, −7.078650739010602, −6.399205041275680, −5.893023967272364, −5.531504423643440, −4.672103698026508, −3.997158988048443, −3.807959415503246, −2.904364773311531, −2.480317925082311, −1.790617353088887, −1.009670167487658, −0.4130811147205147,
0.4130811147205147, 1.009670167487658, 1.790617353088887, 2.480317925082311, 2.904364773311531, 3.807959415503246, 3.997158988048443, 4.672103698026508, 5.531504423643440, 5.893023967272364, 6.399205041275680, 7.078650739010602, 7.289334264780924, 8.032211729603074, 8.327775787467330, 8.939112837005949, 9.243499155766077, 9.813059071371769, 10.34352460278080, 10.82967764021871, 11.15180683827740, 11.59422607198974, 12.29104682748032, 12.58282435550079, 12.99415061524357
