| L(s) = 1 | + 2.61·2-s + 0.492·3-s + 4.82·4-s + 5-s + 1.28·6-s + 3.47·7-s + 7.37·8-s − 2.75·9-s + 2.61·10-s + 11-s + 2.37·12-s − 0.698·13-s + 9.06·14-s + 0.492·15-s + 9.61·16-s + 6.56·17-s − 7.20·18-s + 3.10·19-s + 4.82·20-s + 1.70·21-s + 2.61·22-s − 3.07·23-s + 3.62·24-s + 25-s − 1.82·26-s − 2.83·27-s + 16.7·28-s + ⋯ |
| L(s) = 1 | + 1.84·2-s + 0.284·3-s + 2.41·4-s + 0.447·5-s + 0.524·6-s + 1.31·7-s + 2.60·8-s − 0.919·9-s + 0.825·10-s + 0.301·11-s + 0.685·12-s − 0.193·13-s + 2.42·14-s + 0.127·15-s + 2.40·16-s + 1.59·17-s − 1.69·18-s + 0.711·19-s + 1.07·20-s + 0.372·21-s + 0.556·22-s − 0.641·23-s + 0.740·24-s + 0.200·25-s − 0.357·26-s − 0.545·27-s + 3.16·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.553402377\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.553402377\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| good | 2 | \( 1 - 2.61T + 2T^{2} \) |
| 3 | \( 1 - 0.492T + 3T^{2} \) |
| 7 | \( 1 - 3.47T + 7T^{2} \) |
| 13 | \( 1 + 0.698T + 13T^{2} \) |
| 17 | \( 1 - 6.56T + 17T^{2} \) |
| 19 | \( 1 - 3.10T + 19T^{2} \) |
| 23 | \( 1 + 3.07T + 23T^{2} \) |
| 29 | \( 1 + 4.74T + 29T^{2} \) |
| 31 | \( 1 + 8.90T + 31T^{2} \) |
| 37 | \( 1 + 1.42T + 37T^{2} \) |
| 41 | \( 1 + 5.91T + 41T^{2} \) |
| 43 | \( 1 + 10.0T + 43T^{2} \) |
| 47 | \( 1 - 0.200T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 + 7.84T + 59T^{2} \) |
| 61 | \( 1 + 8.96T + 61T^{2} \) |
| 67 | \( 1 - 13.0T + 67T^{2} \) |
| 71 | \( 1 + 4.05T + 71T^{2} \) |
| 79 | \( 1 - 15.0T + 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 - 13.7T + 97T^{2} \) |
| 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
−8.097527791371719315593830721242, −7.61605187740950412096744014272, −6.78800958411912719886075285840, −5.73865344907480685476372432032, −5.43185446663838081540266789916, −4.87939007442420308118889335977, −3.73105646416311990019974138693, −3.28822347927592567070381431097, −2.18622396018091752165090325237, −1.54059595468405124647464467684,
1.54059595468405124647464467684, 2.18622396018091752165090325237, 3.28822347927592567070381431097, 3.73105646416311990019974138693, 4.87939007442420308118889335977, 5.43185446663838081540266789916, 5.73865344907480685476372432032, 6.78800958411912719886075285840, 7.61605187740950412096744014272, 8.097527791371719315593830721242
