| L(s) = 1 | + 2-s + 1.56·3-s + 4-s + 5-s + 1.56·6-s − 3.12·7-s + 8-s − 0.561·9-s + 10-s − 4·11-s + 1.56·12-s + 0.438·13-s − 3.12·14-s + 1.56·15-s + 16-s + 17-s − 0.561·18-s + 1.56·19-s + 20-s − 4.87·21-s − 4·22-s + 3.12·23-s + 1.56·24-s + 25-s + 0.438·26-s − 5.56·27-s − 3.12·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.901·3-s + 0.5·4-s + 0.447·5-s + 0.637·6-s − 1.18·7-s + 0.353·8-s − 0.187·9-s + 0.316·10-s − 1.20·11-s + 0.450·12-s + 0.121·13-s − 0.834·14-s + 0.403·15-s + 0.250·16-s + 0.242·17-s − 0.132·18-s + 0.358·19-s + 0.223·20-s − 1.06·21-s − 0.852·22-s + 0.651·23-s + 0.318·24-s + 0.200·25-s + 0.0859·26-s − 1.07·27-s − 0.590·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.971191833\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.971191833\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - 1.56T + 3T^{2} \) |
| 7 | \( 1 + 3.12T + 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 0.438T + 13T^{2} \) |
| 19 | \( 1 - 1.56T + 19T^{2} \) |
| 23 | \( 1 - 3.12T + 23T^{2} \) |
| 29 | \( 1 - 6.68T + 29T^{2} \) |
| 31 | \( 1 + 2.43T + 31T^{2} \) |
| 37 | \( 1 - 1.12T + 37T^{2} \) |
| 41 | \( 1 + 12.2T + 41T^{2} \) |
| 43 | \( 1 - 7.12T + 43T^{2} \) |
| 47 | \( 1 - 2.43T + 47T^{2} \) |
| 53 | \( 1 - 3.56T + 53T^{2} \) |
| 59 | \( 1 + 12.6T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 + 0.876T + 67T^{2} \) |
| 71 | \( 1 + 16.6T + 71T^{2} \) |
| 73 | \( 1 - 13.8T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 - 2.68T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.13877933635486059572757260188, −12.10544810541229785350091760547, −10.68506036829011038462301877555, −9.784030126246858738436207990935, −8.711738385693752819143107421098, −7.52351833611557182118683950689, −6.29590725444563709729946214901, −5.17293503921575241435705278419, −3.40453406978190707729069353761, −2.58583688898183257087516876630,
2.58583688898183257087516876630, 3.40453406978190707729069353761, 5.17293503921575241435705278419, 6.29590725444563709729946214901, 7.52351833611557182118683950689, 8.711738385693752819143107421098, 9.784030126246858738436207990935, 10.68506036829011038462301877555, 12.10544810541229785350091760547, 13.13877933635486059572757260188
