| L(s) = 1 | − 2-s − 4-s − 5-s − 7-s + 3·8-s + 10-s − 4·11-s + 2·13-s + 14-s − 16-s − 4·19-s + 20-s + 4·22-s + 4·23-s + 25-s − 2·26-s + 28-s + 10·29-s − 4·31-s − 5·32-s + 35-s + 10·37-s + 4·38-s − 3·40-s + 6·41-s + 4·43-s + 4·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.447·5-s − 0.377·7-s + 1.06·8-s + 0.316·10-s − 1.20·11-s + 0.554·13-s + 0.267·14-s − 1/4·16-s − 0.917·19-s + 0.223·20-s + 0.852·22-s + 0.834·23-s + 1/5·25-s − 0.392·26-s + 0.188·28-s + 1.85·29-s − 0.718·31-s − 0.883·32-s + 0.169·35-s + 1.64·37-s + 0.648·38-s − 0.474·40-s + 0.937·41-s + 0.609·43-s + 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9199854178\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9199854178\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.78967589521223, −13.17832707931947, −12.89760264909922, −12.64905261782508, −11.80457142635009, −11.12948717007135, −10.88878718046533, −10.22581528639728, −10.01412869514220, −9.284861650328033, −8.775544932971569, −8.397472424696349, −7.959506036363867, −7.382051973337847, −6.958264080381564, −6.144492972972249, −5.722094658220533, −4.926705299610822, −4.448970276281815, −4.076787721083371, −3.107845546863220, −2.760875637654062, −1.868073984346464, −0.9650601961618833, −0.4357688975093844,
0.4357688975093844, 0.9650601961618833, 1.868073984346464, 2.760875637654062, 3.107845546863220, 4.076787721083371, 4.448970276281815, 4.926705299610822, 5.722094658220533, 6.144492972972249, 6.958264080381564, 7.382051973337847, 7.959506036363867, 8.397472424696349, 8.775544932971569, 9.284861650328033, 10.01412869514220, 10.22581528639728, 10.88878718046533, 11.12948717007135, 11.80457142635009, 12.64905261782508, 12.89760264909922, 13.17832707931947, 13.78967589521223
