L(s) = 1 | − 2.76·2-s + 3-s + 5.62·4-s − 0.677·5-s − 2.76·6-s + 3.14·7-s − 9.99·8-s + 9-s + 1.86·10-s − 2.07·11-s + 5.62·12-s + 5.11·13-s − 8.67·14-s − 0.677·15-s + 16.3·16-s − 17-s − 2.76·18-s + 1.32·19-s − 3.80·20-s + 3.14·21-s + 5.72·22-s + 1.26·23-s − 9.99·24-s − 4.54·25-s − 14.1·26-s + 27-s + 17.6·28-s + ⋯ |
L(s) = 1 | − 1.95·2-s + 0.577·3-s + 2.81·4-s − 0.302·5-s − 1.12·6-s + 1.18·7-s − 3.53·8-s + 0.333·9-s + 0.591·10-s − 0.625·11-s + 1.62·12-s + 1.41·13-s − 2.31·14-s − 0.174·15-s + 4.08·16-s − 0.242·17-s − 0.650·18-s + 0.302·19-s − 0.851·20-s + 0.686·21-s + 1.22·22-s + 0.263·23-s − 2.04·24-s − 0.908·25-s − 2.77·26-s + 0.192·27-s + 3.33·28-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 | \( 1 + 2.76T + 2T^{2} \) |
| 5 | \( 1 + 0.677T + 5T^{2} \) |
| 7 | \( 1 - 3.14T + 7T^{2} \) |
| 11 | \( 1 + 2.07T + 11T^{2} \) |
| 13 | \( 1 - 5.11T + 13T^{2} \) |
| 19 | \( 1 - 1.32T + 19T^{2} \) |
| 23 | \( 1 - 1.26T + 23T^{2} \) |
| 29 | \( 1 - 2.79T + 29T^{2} \) |
| 31 | \( 1 + 1.04T + 31T^{2} \) |
| 37 | \( 1 - 1.14T + 37T^{2} \) |
| 41 | \( 1 + 8.84T + 41T^{2} \) |
| 43 | \( 1 + 12.3T + 43T^{2} \) |
| 47 | \( 1 - 2.89T + 47T^{2} \) |
| 53 | \( 1 + 6.36T + 53T^{2} \) |
| 59 | \( 1 + 4.43T + 59T^{2} \) |
| 61 | \( 1 + 14.2T + 61T^{2} \) |
| 67 | \( 1 + 6.45T + 67T^{2} \) |
| 71 | \( 1 + 13.7T + 71T^{2} \) |
| 73 | \( 1 + 0.815T + 73T^{2} \) |
| 79 | \( 1 + 5.63T + 79T^{2} \) |
| 83 | \( 1 - 3.91T + 83T^{2} \) |
| 89 | \( 1 - 6.79T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−7.88578562528193018332625221567, −7.20292499739211738812536628208, −6.44735398679393799179168029636, −5.69569611272620702619351148638, −4.66733793595197221640642088172, −3.49372148376215033351693216852, −2.82366069794294298632799116102, −1.70069029315566022738981080928, −1.40658309437981031733246350617, 0,
1.40658309437981031733246350617, 1.70069029315566022738981080928, 2.82366069794294298632799116102, 3.49372148376215033351693216852, 4.66733793595197221640642088172, 5.69569611272620702619351148638, 6.44735398679393799179168029636, 7.20292499739211738812536628208, 7.88578562528193018332625221567