L(s) = 1 | − 2-s + 0.317·3-s + 4-s − 0.609·5-s − 0.317·6-s − 2.41·7-s − 8-s − 2.89·9-s + 0.609·10-s + 5.46·11-s + 0.317·12-s − 5.32·13-s + 2.41·14-s − 0.193·15-s + 16-s + 1.32·17-s + 2.89·18-s + 0.523·19-s − 0.609·20-s − 0.766·21-s − 5.46·22-s + 2.78·23-s − 0.317·24-s − 4.62·25-s + 5.32·26-s − 1.87·27-s − 2.41·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.183·3-s + 0.5·4-s − 0.272·5-s − 0.129·6-s − 0.913·7-s − 0.353·8-s − 0.966·9-s + 0.192·10-s + 1.64·11-s + 0.0915·12-s − 1.47·13-s + 0.646·14-s − 0.0499·15-s + 0.250·16-s + 0.320·17-s + 0.683·18-s + 0.120·19-s − 0.136·20-s − 0.167·21-s − 1.16·22-s + 0.580·23-s − 0.0647·24-s − 0.925·25-s + 1.04·26-s − 0.360·27-s − 0.456·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8546017580\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8546017580\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 2017 | \( 1 - T \) |
good | 3 | \( 1 - 0.317T + 3T^{2} \) |
| 5 | \( 1 + 0.609T + 5T^{2} \) |
| 7 | \( 1 + 2.41T + 7T^{2} \) |
| 11 | \( 1 - 5.46T + 11T^{2} \) |
| 13 | \( 1 + 5.32T + 13T^{2} \) |
| 17 | \( 1 - 1.32T + 17T^{2} \) |
| 19 | \( 1 - 0.523T + 19T^{2} \) |
| 23 | \( 1 - 2.78T + 23T^{2} \) |
| 29 | \( 1 - 6.19T + 29T^{2} \) |
| 31 | \( 1 + 7.39T + 31T^{2} \) |
| 37 | \( 1 - 0.174T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 - 5.74T + 47T^{2} \) |
| 53 | \( 1 + 8.09T + 53T^{2} \) |
| 59 | \( 1 - 6.72T + 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 + 15.1T + 67T^{2} \) |
| 71 | \( 1 + 3.88T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 + 2.99T + 83T^{2} \) |
| 89 | \( 1 + 2.91T + 89T^{2} \) |
| 97 | \( 1 - 14.8T + 97T^{2} \) |
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
−8.656710501650470251893564125074, −7.65761292341099973085563972123, −7.12099454824980577620264281675, −6.36433922732891079246279151124, −5.71870288119135026673770626240, −4.62067449976762009089437998947, −3.57180468377275813326561171087, −2.95704659201064588930141332423, −1.93268536613653003240685657874, −0.56257102983720194342085889838,
0.56257102983720194342085889838, 1.93268536613653003240685657874, 2.95704659201064588930141332423, 3.57180468377275813326561171087, 4.62067449976762009089437998947, 5.71870288119135026673770626240, 6.36433922732891079246279151124, 7.12099454824980577620264281675, 7.65761292341099973085563972123, 8.656710501650470251893564125074