L(s) = 1 | + 0.365·2-s + 3-s − 1.86·4-s − 3.97·5-s + 0.365·6-s + 3.07·7-s − 1.41·8-s + 9-s − 1.45·10-s + 0.220·11-s − 1.86·12-s − 3.98·13-s + 1.12·14-s − 3.97·15-s + 3.21·16-s + 17-s + 0.365·18-s − 0.328·19-s + 7.42·20-s + 3.07·21-s + 0.0804·22-s − 2.08·23-s − 1.41·24-s + 10.8·25-s − 1.45·26-s + 27-s − 5.74·28-s + ⋯ |
L(s) = 1 | + 0.258·2-s + 0.577·3-s − 0.933·4-s − 1.77·5-s + 0.149·6-s + 1.16·7-s − 0.499·8-s + 0.333·9-s − 0.460·10-s + 0.0663·11-s − 0.538·12-s − 1.10·13-s + 0.300·14-s − 1.02·15-s + 0.803·16-s + 0.242·17-s + 0.0861·18-s − 0.0753·19-s + 1.66·20-s + 0.671·21-s + 0.0171·22-s − 0.433·23-s − 0.288·24-s + 2.16·25-s − 0.285·26-s + 0.192·27-s − 1.08·28-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 | \( 1 - 0.365T + 2T^{2} \) |
| 5 | \( 1 + 3.97T + 5T^{2} \) |
| 7 | \( 1 - 3.07T + 7T^{2} \) |
| 11 | \( 1 - 0.220T + 11T^{2} \) |
| 13 | \( 1 + 3.98T + 13T^{2} \) |
| 19 | \( 1 + 0.328T + 19T^{2} \) |
| 23 | \( 1 + 2.08T + 23T^{2} \) |
| 29 | \( 1 + 1.31T + 29T^{2} \) |
| 31 | \( 1 + 0.177T + 31T^{2} \) |
| 37 | \( 1 - 8.74T + 37T^{2} \) |
| 41 | \( 1 + 1.77T + 41T^{2} \) |
| 43 | \( 1 - 7.05T + 43T^{2} \) |
| 47 | \( 1 + 3.10T + 47T^{2} \) |
| 53 | \( 1 + 7.51T + 53T^{2} \) |
| 59 | \( 1 - 1.69T + 59T^{2} \) |
| 61 | \( 1 - 6.89T + 61T^{2} \) |
| 67 | \( 1 + 2.47T + 67T^{2} \) |
| 71 | \( 1 + 0.954T + 71T^{2} \) |
| 73 | \( 1 + 7.59T + 73T^{2} \) |
| 79 | \( 1 - 1.89T + 79T^{2} \) |
| 83 | \( 1 - 1.21T + 83T^{2} \) |
| 89 | \( 1 + 10.2T + 89T^{2} \) |
| 97 | \( 1 - 6.57T + 97T^{2} \) |
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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
−7.68568365334936275621550851859, −7.20362345140698948454840752205, −5.99734774730694014562543745823, −4.91076014781893933866922705726, −4.64461658789017407881745791022, −3.99669751799542938008999158527, −3.35580589721727613525794933158, −2.42458639839715162500143857584, −1.09491862335192492952708883871, 0,
1.09491862335192492952708883871, 2.42458639839715162500143857584, 3.35580589721727613525794933158, 3.99669751799542938008999158527, 4.64461658789017407881745791022, 4.91076014781893933866922705726, 5.99734774730694014562543745823, 7.20362345140698948454840752205, 7.68568365334936275621550851859