L(s) = 1 | − 0.412·2-s + 1.67·3-s − 1.83·4-s − 5-s − 0.691·6-s + 0.0703·7-s + 1.57·8-s − 0.184·9-s + 0.412·10-s − 11-s − 3.07·12-s + 1.09·13-s − 0.0289·14-s − 1.67·15-s + 3.00·16-s − 2.37·17-s + 0.0758·18-s − 19-s + 1.83·20-s + 0.117·21-s + 0.412·22-s − 2.83·23-s + 2.64·24-s + 25-s − 0.449·26-s − 5.34·27-s − 0.128·28-s + ⋯ |
L(s) = 1 | − 0.291·2-s + 0.968·3-s − 0.915·4-s − 0.447·5-s − 0.282·6-s + 0.0265·7-s + 0.558·8-s − 0.0613·9-s + 0.130·10-s − 0.301·11-s − 0.886·12-s + 0.302·13-s − 0.00774·14-s − 0.433·15-s + 0.752·16-s − 0.577·17-s + 0.0178·18-s − 0.229·19-s + 0.409·20-s + 0.0257·21-s + 0.0878·22-s − 0.590·23-s + 0.540·24-s + 0.200·25-s − 0.0881·26-s − 1.02·27-s − 0.0243·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
good | 2 | \( 1 + 0.412T + 2T^{2} \) |
| 3 | \( 1 - 1.67T + 3T^{2} \) |
| 7 | \( 1 - 0.0703T + 7T^{2} \) |
| 13 | \( 1 - 1.09T + 13T^{2} \) |
| 17 | \( 1 + 2.37T + 17T^{2} \) |
| 23 | \( 1 + 2.83T + 23T^{2} \) |
| 29 | \( 1 + 9.85T + 29T^{2} \) |
| 31 | \( 1 - 1.51T + 31T^{2} \) |
| 37 | \( 1 + 1.38T + 37T^{2} \) |
| 41 | \( 1 - 2.59T + 41T^{2} \) |
| 43 | \( 1 + 5.63T + 43T^{2} \) |
| 47 | \( 1 + 4.95T + 47T^{2} \) |
| 53 | \( 1 + 3.23T + 53T^{2} \) |
| 59 | \( 1 - 5.63T + 59T^{2} \) |
| 61 | \( 1 - 1.13T + 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 + 3.37T + 71T^{2} \) |
| 73 | \( 1 + 4.54T + 73T^{2} \) |
| 79 | \( 1 - 2.22T + 79T^{2} \) |
| 83 | \( 1 - 2.48T + 83T^{2} \) |
| 89 | \( 1 - 6.45T + 89T^{2} \) |
| 97 | \( 1 - 9.43T + 97T^{2} \) |
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
−9.285585871062046068719510638172, −8.685692888397283206738086163140, −8.048044699165797124862163227704, −7.41573262184427871949513638976, −6.07334005372891380115899414012, −4.98574385124782228315384122898, −4.00307030489092691870585099779, −3.25184117695079199248778051745, −1.88449877567300572374722602720, 0,
1.88449877567300572374722602720, 3.25184117695079199248778051745, 4.00307030489092691870585099779, 4.98574385124782228315384122898, 6.07334005372891380115899414012, 7.41573262184427871949513638976, 8.048044699165797124862163227704, 8.685692888397283206738086163140, 9.285585871062046068719510638172