| L(s) = 1 | + 2-s − 2.56·3-s + 4-s − 3.99·5-s − 2.56·6-s + 2.48·7-s + 8-s + 3.59·9-s − 3.99·10-s + 3.77·11-s − 2.56·12-s + 5.52·13-s + 2.48·14-s + 10.2·15-s + 16-s + 6.23·17-s + 3.59·18-s + 4.57·19-s − 3.99·20-s − 6.39·21-s + 3.77·22-s − 2.19·23-s − 2.56·24-s + 10.9·25-s + 5.52·26-s − 1.53·27-s + 2.48·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.48·3-s + 0.5·4-s − 1.78·5-s − 1.04·6-s + 0.940·7-s + 0.353·8-s + 1.19·9-s − 1.26·10-s + 1.13·11-s − 0.741·12-s + 1.53·13-s + 0.665·14-s + 2.65·15-s + 0.250·16-s + 1.51·17-s + 0.848·18-s + 1.05·19-s − 0.894·20-s − 1.39·21-s + 0.804·22-s − 0.457·23-s − 0.524·24-s + 2.19·25-s + 1.08·26-s − 0.295·27-s + 0.470·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.959621457\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.959621457\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 - T \) |
| good | 3 | \( 1 + 2.56T + 3T^{2} \) |
| 5 | \( 1 + 3.99T + 5T^{2} \) |
| 7 | \( 1 - 2.48T + 7T^{2} \) |
| 11 | \( 1 - 3.77T + 11T^{2} \) |
| 13 | \( 1 - 5.52T + 13T^{2} \) |
| 17 | \( 1 - 6.23T + 17T^{2} \) |
| 19 | \( 1 - 4.57T + 19T^{2} \) |
| 23 | \( 1 + 2.19T + 23T^{2} \) |
| 29 | \( 1 + 5.87T + 29T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 + 9.39T + 41T^{2} \) |
| 43 | \( 1 - 7.43T + 43T^{2} \) |
| 47 | \( 1 + 3.21T + 47T^{2} \) |
| 53 | \( 1 + 4.47T + 53T^{2} \) |
| 59 | \( 1 + 0.0676T + 59T^{2} \) |
| 61 | \( 1 - 1.67T + 61T^{2} \) |
| 67 | \( 1 + 12.0T + 67T^{2} \) |
| 71 | \( 1 - 4.55T + 71T^{2} \) |
| 73 | \( 1 - 9.15T + 73T^{2} \) |
| 79 | \( 1 + 9.61T + 79T^{2} \) |
| 83 | \( 1 - 13.4T + 83T^{2} \) |
| 89 | \( 1 - 17.5T + 89T^{2} \) |
| 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
−7.79127858378884708069293803399, −7.39625309609692916728908002018, −6.42924785072411953674022424357, −5.92147575876362436008610458948, −5.12659579479079448294858956967, −4.50440087190129689096973727387, −3.76840575609864746564493352386, −3.35656777590985841379838630779, −1.40276843242009153681281426916, −0.836328878379418462585627158216,
0.836328878379418462585627158216, 1.40276843242009153681281426916, 3.35656777590985841379838630779, 3.76840575609864746564493352386, 4.50440087190129689096973727387, 5.12659579479079448294858956967, 5.92147575876362436008610458948, 6.42924785072411953674022424357, 7.39625309609692916728908002018, 7.79127858378884708069293803399
