| L(s) = 1 | − 1.60·2-s − 3-s + 0.590·4-s + 2.17·5-s + 1.60·6-s + 3.27·7-s + 2.26·8-s + 9-s − 3.49·10-s + 0.0669·11-s − 0.590·12-s + 0.443·13-s − 5.27·14-s − 2.17·15-s − 4.83·16-s + 17-s − 1.60·18-s − 3.50·19-s + 1.28·20-s − 3.27·21-s − 0.107·22-s − 4.40·23-s − 2.26·24-s − 0.286·25-s − 0.713·26-s − 27-s + 1.93·28-s + ⋯ |
| L(s) = 1 | − 1.13·2-s − 0.577·3-s + 0.295·4-s + 0.970·5-s + 0.657·6-s + 1.23·7-s + 0.802·8-s + 0.333·9-s − 1.10·10-s + 0.0201·11-s − 0.170·12-s + 0.122·13-s − 1.40·14-s − 0.560·15-s − 1.20·16-s + 0.242·17-s − 0.379·18-s − 0.803·19-s + 0.286·20-s − 0.714·21-s − 0.0229·22-s − 0.918·23-s − 0.463·24-s − 0.0573·25-s − 0.139·26-s − 0.192·27-s + 0.365·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 - T \) |
| good | 2 | \( 1 + 1.60T + 2T^{2} \) |
| 5 | \( 1 - 2.17T + 5T^{2} \) |
| 7 | \( 1 - 3.27T + 7T^{2} \) |
| 11 | \( 1 - 0.0669T + 11T^{2} \) |
| 13 | \( 1 - 0.443T + 13T^{2} \) |
| 19 | \( 1 + 3.50T + 19T^{2} \) |
| 23 | \( 1 + 4.40T + 23T^{2} \) |
| 29 | \( 1 - 8.76T + 29T^{2} \) |
| 31 | \( 1 - 9.92T + 31T^{2} \) |
| 37 | \( 1 + 3.50T + 37T^{2} \) |
| 41 | \( 1 + 8.85T + 41T^{2} \) |
| 43 | \( 1 + 0.643T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 - 7.39T + 59T^{2} \) |
| 61 | \( 1 + 7.94T + 61T^{2} \) |
| 67 | \( 1 + 15.6T + 67T^{2} \) |
| 71 | \( 1 + 2.66T + 71T^{2} \) |
| 73 | \( 1 - 11.0T + 73T^{2} \) |
| 79 | \( 1 + 6.51T + 79T^{2} \) |
| 83 | \( 1 + 3.88T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 + 10.3T + 97T^{2} \) |
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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
−7.82817799163735674368947506687, −6.69552795889610945807665620336, −6.35692178512042048479916588704, −5.36288649743904496291828610614, −4.76406851750512718030723920978, −4.20534291383606006906266046089, −2.73458600721658073594612317279, −1.66087478517281213383312689858, −1.37885569398835792568496535668, 0,
1.37885569398835792568496535668, 1.66087478517281213383312689858, 2.73458600721658073594612317279, 4.20534291383606006906266046089, 4.76406851750512718030723920978, 5.36288649743904496291828610614, 6.35692178512042048479916588704, 6.69552795889610945807665620336, 7.82817799163735674368947506687
