L(s) = 1 | + 0.723·2-s − 0.489·3-s − 1.47·4-s − 2.47·5-s − 0.353·6-s + 1.15·7-s − 2.51·8-s − 2.76·9-s − 1.78·10-s − 11-s + 0.722·12-s − 2.87·13-s + 0.834·14-s + 1.20·15-s + 1.13·16-s + 17-s − 1.99·18-s − 0.684·19-s + 3.65·20-s − 0.564·21-s − 0.723·22-s − 6.52·23-s + 1.22·24-s + 1.11·25-s − 2.07·26-s + 2.81·27-s − 1.70·28-s + ⋯ |
L(s) = 1 | + 0.511·2-s − 0.282·3-s − 0.738·4-s − 1.10·5-s − 0.144·6-s + 0.436·7-s − 0.889·8-s − 0.920·9-s − 0.565·10-s − 0.301·11-s + 0.208·12-s − 0.796·13-s + 0.223·14-s + 0.312·15-s + 0.283·16-s + 0.242·17-s − 0.470·18-s − 0.157·19-s + 0.816·20-s − 0.123·21-s − 0.154·22-s − 1.36·23-s + 0.251·24-s + 0.222·25-s − 0.407·26-s + 0.542·27-s − 0.321·28-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
\(L(1)\) |
\(\approx\) |
\(0.07072906928\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.07072906928\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.723T + 2T^{2} \) |
| 3 | \( 1 + 0.489T + 3T^{2} \) |
| 5 | \( 1 + 2.47T + 5T^{2} \) |
| 7 | \( 1 - 1.15T + 7T^{2} \) |
| 13 | \( 1 + 2.87T + 13T^{2} \) |
| 19 | \( 1 + 0.684T + 19T^{2} \) |
| 23 | \( 1 + 6.52T + 23T^{2} \) |
| 29 | \( 1 - 0.102T + 29T^{2} \) |
| 31 | \( 1 + 5.40T + 31T^{2} \) |
| 37 | \( 1 + 8.58T + 37T^{2} \) |
| 41 | \( 1 + 5.75T + 41T^{2} \) |
| 47 | \( 1 + 8.93T + 47T^{2} \) |
| 53 | \( 1 + 5.36T + 53T^{2} \) |
| 59 | \( 1 - 5.09T + 59T^{2} \) |
| 61 | \( 1 - 1.42T + 61T^{2} \) |
| 67 | \( 1 + 1.01T + 67T^{2} \) |
| 71 | \( 1 + 15.4T + 71T^{2} \) |
| 73 | \( 1 + 15.5T + 73T^{2} \) |
| 79 | \( 1 + 4.14T + 79T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 - 16.6T + 89T^{2} \) |
| 97 | \( 1 - 4.43T + 97T^{2} \) |
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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
−7.899667618355556946140247925769, −7.25635149495947448915066838600, −6.25902995134975340612210853548, −5.55490087429576482346690315537, −4.96814846571916363992091044294, −4.39057450720407484405001122784, −3.56932177368940376593527252833, −3.03029445870740440904775415214, −1.80418627828934060755636464076, −0.11839996022844399864885554193,
0.11839996022844399864885554193, 1.80418627828934060755636464076, 3.03029445870740440904775415214, 3.56932177368940376593527252833, 4.39057450720407484405001122784, 4.96814846571916363992091044294, 5.55490087429576482346690315537, 6.25902995134975340612210853548, 7.25635149495947448915066838600, 7.899667618355556946140247925769