| L(s) = 1 | − 0.930·2-s − 1.13·4-s + 5-s − 1.89·7-s + 2.91·8-s − 0.930·10-s − 1.68·11-s + 3.98·13-s + 1.76·14-s − 0.448·16-s + 5.63·17-s − 4.84·19-s − 1.13·20-s + 1.56·22-s − 3.34·23-s + 25-s − 3.70·26-s + 2.14·28-s − 10.4·29-s − 6.89·31-s − 5.41·32-s − 5.24·34-s − 1.89·35-s + 9.77·37-s + 4.51·38-s + 2.91·40-s − 0.369·41-s + ⋯ |
| L(s) = 1 | − 0.658·2-s − 0.566·4-s + 0.447·5-s − 0.715·7-s + 1.03·8-s − 0.294·10-s − 0.507·11-s + 1.10·13-s + 0.471·14-s − 0.112·16-s + 1.36·17-s − 1.11·19-s − 0.253·20-s + 0.334·22-s − 0.696·23-s + 0.200·25-s − 0.726·26-s + 0.405·28-s − 1.94·29-s − 1.23·31-s − 0.957·32-s − 0.899·34-s − 0.319·35-s + 1.60·37-s + 0.732·38-s + 0.461·40-s − 0.0576·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 89 | \( 1 + T \) |
| good | 2 | \( 1 + 0.930T + 2T^{2} \) |
| 7 | \( 1 + 1.89T + 7T^{2} \) |
| 11 | \( 1 + 1.68T + 11T^{2} \) |
| 13 | \( 1 - 3.98T + 13T^{2} \) |
| 17 | \( 1 - 5.63T + 17T^{2} \) |
| 19 | \( 1 + 4.84T + 19T^{2} \) |
| 23 | \( 1 + 3.34T + 23T^{2} \) |
| 29 | \( 1 + 10.4T + 29T^{2} \) |
| 31 | \( 1 + 6.89T + 31T^{2} \) |
| 37 | \( 1 - 9.77T + 37T^{2} \) |
| 41 | \( 1 + 0.369T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 + 4.23T + 47T^{2} \) |
| 53 | \( 1 + 4.47T + 53T^{2} \) |
| 59 | \( 1 + 1.24T + 59T^{2} \) |
| 61 | \( 1 - 8.22T + 61T^{2} \) |
| 67 | \( 1 + 1.79T + 67T^{2} \) |
| 71 | \( 1 - 4.91T + 71T^{2} \) |
| 73 | \( 1 - 16.2T + 73T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 + 8.95T + 83T^{2} \) |
| 97 | \( 1 + 18.6T + 97T^{2} \) |
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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
−7.994403606668973896265104717269, −7.73223982954466474520907246277, −6.59385094370263100767801258877, −5.82830085102317993490555541724, −5.29469550865108410446448337053, −4.05414753565760281010717529394, −3.57304814906300969163983649677, −2.28180399093890251328658411391, −1.23295329404923260085120815991, 0,
1.23295329404923260085120815991, 2.28180399093890251328658411391, 3.57304814906300969163983649677, 4.05414753565760281010717529394, 5.29469550865108410446448337053, 5.82830085102317993490555541724, 6.59385094370263100767801258877, 7.73223982954466474520907246277, 7.994403606668973896265104717269
