| L(s) = 1 | + 0.957·2-s + 0.564·3-s − 1.08·4-s − 4.10·5-s + 0.540·6-s + 4.97·7-s − 2.95·8-s − 2.68·9-s − 3.92·10-s + 0.0769·11-s − 0.611·12-s − 3.33·13-s + 4.76·14-s − 2.31·15-s − 0.660·16-s − 5.96·17-s − 2.56·18-s − 7.28·19-s + 4.44·20-s + 2.80·21-s + 0.0736·22-s + 2.10·23-s − 1.66·24-s + 11.8·25-s − 3.18·26-s − 3.20·27-s − 5.38·28-s + ⋯ |
| L(s) = 1 | + 0.677·2-s + 0.325·3-s − 0.541·4-s − 1.83·5-s + 0.220·6-s + 1.87·7-s − 1.04·8-s − 0.893·9-s − 1.24·10-s + 0.0232·11-s − 0.176·12-s − 0.923·13-s + 1.27·14-s − 0.598·15-s − 0.165·16-s − 1.44·17-s − 0.605·18-s − 1.67·19-s + 0.993·20-s + 0.612·21-s + 0.0157·22-s + 0.439·23-s − 0.340·24-s + 2.36·25-s − 0.625·26-s − 0.617·27-s − 1.01·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 547 | \( 1 + T \) |
| good | 2 | \( 1 - 0.957T + 2T^{2} \) |
| 3 | \( 1 - 0.564T + 3T^{2} \) |
| 5 | \( 1 + 4.10T + 5T^{2} \) |
| 7 | \( 1 - 4.97T + 7T^{2} \) |
| 11 | \( 1 - 0.0769T + 11T^{2} \) |
| 13 | \( 1 + 3.33T + 13T^{2} \) |
| 17 | \( 1 + 5.96T + 17T^{2} \) |
| 19 | \( 1 + 7.28T + 19T^{2} \) |
| 23 | \( 1 - 2.10T + 23T^{2} \) |
| 29 | \( 1 - 3.24T + 29T^{2} \) |
| 31 | \( 1 + 6.20T + 31T^{2} \) |
| 37 | \( 1 - 2.16T + 37T^{2} \) |
| 41 | \( 1 + 5.00T + 41T^{2} \) |
| 43 | \( 1 + 1.06T + 43T^{2} \) |
| 47 | \( 1 - 5.86T + 47T^{2} \) |
| 53 | \( 1 + 9.01T + 53T^{2} \) |
| 59 | \( 1 + 4.47T + 59T^{2} \) |
| 61 | \( 1 - 4.58T + 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 - 3.35T + 71T^{2} \) |
| 73 | \( 1 - 14.7T + 73T^{2} \) |
| 79 | \( 1 + 2.00T + 79T^{2} \) |
| 83 | \( 1 + 8.56T + 83T^{2} \) |
| 89 | \( 1 + 8.84T + 89T^{2} \) |
| 97 | \( 1 + 0.394T + 97T^{2} \) |
| 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
−10.94594478682777794497401516920, −9.033700097256866180088474936688, −8.392306400039972474473329168399, −7.997953281462303090873540626833, −6.82003534028774920771781841890, −5.18398172400347214682669358527, −4.54155523371723530944553053275, −3.90178492471291906406348825114, −2.47958008694092344299158262619, 0,
2.47958008694092344299158262619, 3.90178492471291906406348825114, 4.54155523371723530944553053275, 5.18398172400347214682669358527, 6.82003534028774920771781841890, 7.997953281462303090873540626833, 8.392306400039972474473329168399, 9.033700097256866180088474936688, 10.94594478682777794497401516920
