| L(s) = 1 | + 2.52·2-s + 3-s + 4.39·4-s − 3.06·5-s + 2.52·6-s − 1.89·7-s + 6.05·8-s + 9-s − 7.74·10-s − 6.10·11-s + 4.39·12-s + 6.06·13-s − 4.78·14-s − 3.06·15-s + 6.52·16-s − 17-s + 2.52·18-s + 1.41·19-s − 13.4·20-s − 1.89·21-s − 15.4·22-s + 0.697·23-s + 6.05·24-s + 4.38·25-s + 15.3·26-s + 27-s − 8.32·28-s + ⋯ |
| L(s) = 1 | + 1.78·2-s + 0.577·3-s + 2.19·4-s − 1.36·5-s + 1.03·6-s − 0.715·7-s + 2.14·8-s + 0.333·9-s − 2.44·10-s − 1.84·11-s + 1.26·12-s + 1.68·13-s − 1.27·14-s − 0.790·15-s + 1.63·16-s − 0.242·17-s + 0.596·18-s + 0.325·19-s − 3.01·20-s − 0.413·21-s − 3.29·22-s + 0.145·23-s + 1.23·24-s + 0.876·25-s + 3.00·26-s + 0.192·27-s − 1.57·28-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 | \( 1 - 2.52T + 2T^{2} \) |
| 5 | \( 1 + 3.06T + 5T^{2} \) |
| 7 | \( 1 + 1.89T + 7T^{2} \) |
| 11 | \( 1 + 6.10T + 11T^{2} \) |
| 13 | \( 1 - 6.06T + 13T^{2} \) |
| 19 | \( 1 - 1.41T + 19T^{2} \) |
| 23 | \( 1 - 0.697T + 23T^{2} \) |
| 29 | \( 1 + 6.79T + 29T^{2} \) |
| 31 | \( 1 - 0.344T + 31T^{2} \) |
| 37 | \( 1 + 3.08T + 37T^{2} \) |
| 41 | \( 1 + 4.03T + 41T^{2} \) |
| 43 | \( 1 + 0.755T + 43T^{2} \) |
| 47 | \( 1 + 4.00T + 47T^{2} \) |
| 53 | \( 1 + 0.627T + 53T^{2} \) |
| 59 | \( 1 + 7.13T + 59T^{2} \) |
| 61 | \( 1 + 13.8T + 61T^{2} \) |
| 67 | \( 1 - 6.27T + 67T^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 + 0.512T + 73T^{2} \) |
| 79 | \( 1 + 6.80T + 79T^{2} \) |
| 83 | \( 1 + 10.0T + 83T^{2} \) |
| 89 | \( 1 + 6.41T + 89T^{2} \) |
| 97 | \( 1 + 9.88T + 97T^{2} \) |
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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
−7.34726421034283061564138592059, −6.73510921486994063871985117440, −5.92162407673459418666409915304, −5.27972107409217623824346312263, −4.49334024177748624235865560473, −3.80669393160378172630935977924, −3.27082599183941199650132536544, −2.87395247559088418854819902141, −1.71472671797351718229954201881, 0,
1.71472671797351718229954201881, 2.87395247559088418854819902141, 3.27082599183941199650132536544, 3.80669393160378172630935977924, 4.49334024177748624235865560473, 5.27972107409217623824346312263, 5.92162407673459418666409915304, 6.73510921486994063871985117440, 7.34726421034283061564138592059
