L(s) = 1 | + 4.42·2-s + 9.44·3-s + 11.5·4-s − 8.63·5-s + 41.7·6-s + 12.6·7-s + 15.7·8-s + 62.2·9-s − 38.1·10-s + 14.5·11-s + 109.·12-s − 26.6·13-s + 55.9·14-s − 81.5·15-s − 22.7·16-s + 275.·18-s − 125.·19-s − 99.8·20-s + 119.·21-s + 64.2·22-s − 2.31·23-s + 149.·24-s − 50.5·25-s − 117.·26-s + 333.·27-s + 146.·28-s − 21.8·29-s + ⋯ |
L(s) = 1 | + 1.56·2-s + 1.81·3-s + 1.44·4-s − 0.771·5-s + 2.84·6-s + 0.683·7-s + 0.696·8-s + 2.30·9-s − 1.20·10-s + 0.397·11-s + 2.62·12-s − 0.568·13-s + 1.06·14-s − 1.40·15-s − 0.355·16-s + 3.60·18-s − 1.51·19-s − 1.11·20-s + 1.24·21-s + 0.622·22-s − 0.0210·23-s + 1.26·24-s − 0.404·25-s − 0.889·26-s + 2.37·27-s + 0.987·28-s − 0.139·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(7.016487658\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.016487658\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 \) |
good | 2 | \( 1 - 4.42T + 8T^{2} \) |
| 3 | \( 1 - 9.44T + 27T^{2} \) |
| 5 | \( 1 + 8.63T + 125T^{2} \) |
| 7 | \( 1 - 12.6T + 343T^{2} \) |
| 11 | \( 1 - 14.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 26.6T + 2.19e3T^{2} \) |
| 19 | \( 1 + 125.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 2.31T + 1.21e4T^{2} \) |
| 29 | \( 1 + 21.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 323.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 73.2T + 5.06e4T^{2} \) |
| 41 | \( 1 + 179.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 186.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 235.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 200.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 718.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 727.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 76.9T + 3.00e5T^{2} \) |
| 71 | \( 1 + 923.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 820.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 51.0T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.18e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 86.8T + 7.04e5T^{2} \) |
| 97 | \( 1 - 758.T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.83994496938007243824809703777, −10.59727566259860430950339573299, −9.255880147529164461980953078354, −8.289412217798938712708510784607, −7.55218325386562889401970212933, −6.41927456573235425930732721327, −4.61271525754772652377437242932, −4.12963763953552332174551385052, −3.04751035363347074690164446782, −2.02716598979314183859881085614,
2.02716598979314183859881085614, 3.04751035363347074690164446782, 4.12963763953552332174551385052, 4.61271525754772652377437242932, 6.41927456573235425930732721327, 7.55218325386562889401970212933, 8.289412217798938712708510784607, 9.255880147529164461980953078354, 10.59727566259860430950339573299, 11.83994496938007243824809703777