| L(s) = 1 | − 1.90·2-s + 1.99·3-s + 1.61·4-s − 3.13·5-s − 3.79·6-s − 0.279·7-s + 0.740·8-s + 0.990·9-s + 5.95·10-s + 11-s + 3.21·12-s − 1.37·13-s + 0.531·14-s − 6.25·15-s − 4.62·16-s − 17-s − 1.88·18-s + 2.56·19-s − 5.04·20-s − 0.558·21-s − 1.90·22-s − 5.56·23-s + 1.47·24-s + 4.81·25-s + 2.60·26-s − 4.01·27-s − 0.450·28-s + ⋯ |
| L(s) = 1 | − 1.34·2-s + 1.15·3-s + 0.805·4-s − 1.40·5-s − 1.54·6-s − 0.105·7-s + 0.261·8-s + 0.330·9-s + 1.88·10-s + 0.301·11-s + 0.928·12-s − 0.380·13-s + 0.142·14-s − 1.61·15-s − 1.15·16-s − 0.242·17-s − 0.443·18-s + 0.589·19-s − 1.12·20-s − 0.121·21-s − 0.405·22-s − 1.16·23-s + 0.302·24-s + 0.962·25-s + 0.510·26-s − 0.772·27-s − 0.0850·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5780943939\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5780943939\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 + 1.90T + 2T^{2} \) |
| 3 | \( 1 - 1.99T + 3T^{2} \) |
| 5 | \( 1 + 3.13T + 5T^{2} \) |
| 7 | \( 1 + 0.279T + 7T^{2} \) |
| 13 | \( 1 + 1.37T + 13T^{2} \) |
| 19 | \( 1 - 2.56T + 19T^{2} \) |
| 23 | \( 1 + 5.56T + 23T^{2} \) |
| 29 | \( 1 + 5.48T + 29T^{2} \) |
| 31 | \( 1 + 2.40T + 31T^{2} \) |
| 37 | \( 1 + 6.44T + 37T^{2} \) |
| 41 | \( 1 - 5.16T + 41T^{2} \) |
| 47 | \( 1 - 4.29T + 47T^{2} \) |
| 53 | \( 1 - 3.96T + 53T^{2} \) |
| 59 | \( 1 + 0.143T + 59T^{2} \) |
| 61 | \( 1 - 13.9T + 61T^{2} \) |
| 67 | \( 1 + 9.52T + 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 - 2.40T + 79T^{2} \) |
| 83 | \( 1 + 13.6T + 83T^{2} \) |
| 89 | \( 1 - 11.1T + 89T^{2} \) |
| 97 | \( 1 + 7.90T + 97T^{2} \) |
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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
−7.955120424429865366351413606092, −7.39971102164702148829462259667, −7.13364726388347399889473722366, −5.94473097224838088507030831798, −4.83008027836767737875679463875, −3.92680869114413803548341282923, −3.54764964517422457198705667379, −2.49079785756065509870706200844, −1.70623324222801977613183691807, −0.42809809802389799987660426645,
0.42809809802389799987660426645, 1.70623324222801977613183691807, 2.49079785756065509870706200844, 3.54764964517422457198705667379, 3.92680869114413803548341282923, 4.83008027836767737875679463875, 5.94473097224838088507030831798, 7.13364726388347399889473722366, 7.39971102164702148829462259667, 7.955120424429865366351413606092
