| L(s) = 1 | + 2.15·2-s + 2.92·3-s + 2.63·4-s − 5-s + 6.29·6-s − 0.652·7-s + 1.36·8-s + 5.55·9-s − 2.15·10-s + 6.16·11-s + 7.70·12-s + 2.94·13-s − 1.40·14-s − 2.92·15-s − 2.32·16-s − 7.63·17-s + 11.9·18-s − 1.50·19-s − 2.63·20-s − 1.90·21-s + 13.2·22-s + 2.05·23-s + 3.99·24-s + 25-s + 6.34·26-s + 7.46·27-s − 1.71·28-s + ⋯ |
| L(s) = 1 | + 1.52·2-s + 1.68·3-s + 1.31·4-s − 0.447·5-s + 2.57·6-s − 0.246·7-s + 0.482·8-s + 1.85·9-s − 0.680·10-s + 1.85·11-s + 2.22·12-s + 0.816·13-s − 0.375·14-s − 0.755·15-s − 0.582·16-s − 1.85·17-s + 2.81·18-s − 0.345·19-s − 0.589·20-s − 0.416·21-s + 2.82·22-s + 0.429·23-s + 0.814·24-s + 0.200·25-s + 1.24·26-s + 1.43·27-s − 0.324·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.913641677\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.913641677\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 - 2.15T + 2T^{2} \) |
| 3 | \( 1 - 2.92T + 3T^{2} \) |
| 7 | \( 1 + 0.652T + 7T^{2} \) |
| 11 | \( 1 - 6.16T + 11T^{2} \) |
| 13 | \( 1 - 2.94T + 13T^{2} \) |
| 17 | \( 1 + 7.63T + 17T^{2} \) |
| 19 | \( 1 + 1.50T + 19T^{2} \) |
| 23 | \( 1 - 2.05T + 23T^{2} \) |
| 29 | \( 1 - 2.81T + 29T^{2} \) |
| 31 | \( 1 + 5.74T + 31T^{2} \) |
| 37 | \( 1 - 4.92T + 37T^{2} \) |
| 41 | \( 1 + 1.10T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 + 4.74T + 47T^{2} \) |
| 53 | \( 1 + 4.28T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 + 5.04T + 61T^{2} \) |
| 67 | \( 1 + 8.07T + 67T^{2} \) |
| 71 | \( 1 + 2.29T + 71T^{2} \) |
| 73 | \( 1 + 5.46T + 73T^{2} \) |
| 79 | \( 1 + 3.70T + 79T^{2} \) |
| 83 | \( 1 + 6.84T + 83T^{2} \) |
| 89 | \( 1 + 1.06T + 89T^{2} \) |
| 97 | \( 1 - 11.5T + 97T^{2} \) |
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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
−8.867283655740312138809704964476, −8.672654147679981587720628086885, −7.38850716605108642733047308211, −6.68071198517218237426969944990, −6.12010324495678112654292559332, −4.54379756172351398702583223097, −4.10594042117018449078570608356, −3.51292647210520330429045294779, −2.68173703621387504025959458510, −1.65603056583207887522901560113,
1.65603056583207887522901560113, 2.68173703621387504025959458510, 3.51292647210520330429045294779, 4.10594042117018449078570608356, 4.54379756172351398702583223097, 6.12010324495678112654292559332, 6.68071198517218237426969944990, 7.38850716605108642733047308211, 8.672654147679981587720628086885, 8.867283655740312138809704964476
