| L(s) = 1 | + 1.88·2-s + 1.53·4-s + 5-s − 0.422·7-s − 0.869·8-s + 1.88·10-s + 2.50·11-s − 3.68·13-s − 0.794·14-s − 4.71·16-s − 3.71·17-s − 3.03·19-s + 1.53·20-s + 4.70·22-s − 7.79·23-s + 25-s − 6.93·26-s − 0.649·28-s + 0.762·29-s − 7.82·31-s − 7.12·32-s − 6.99·34-s − 0.422·35-s − 2.51·37-s − 5.71·38-s − 0.869·40-s + 7.74·41-s + ⋯ |
| L(s) = 1 | + 1.32·2-s + 0.768·4-s + 0.447·5-s − 0.159·7-s − 0.307·8-s + 0.594·10-s + 0.754·11-s − 1.02·13-s − 0.212·14-s − 1.17·16-s − 0.901·17-s − 0.696·19-s + 0.343·20-s + 1.00·22-s − 1.62·23-s + 0.200·25-s − 1.35·26-s − 0.122·28-s + 0.141·29-s − 1.40·31-s − 1.25·32-s − 1.19·34-s − 0.0713·35-s − 0.413·37-s − 0.926·38-s − 0.137·40-s + 1.20·41-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 | \( 1 - 1.88T + 2T^{2} \) |
| 7 | \( 1 + 0.422T + 7T^{2} \) |
| 11 | \( 1 - 2.50T + 11T^{2} \) |
| 13 | \( 1 + 3.68T + 13T^{2} \) |
| 17 | \( 1 + 3.71T + 17T^{2} \) |
| 19 | \( 1 + 3.03T + 19T^{2} \) |
| 23 | \( 1 + 7.79T + 23T^{2} \) |
| 29 | \( 1 - 0.762T + 29T^{2} \) |
| 31 | \( 1 + 7.82T + 31T^{2} \) |
| 37 | \( 1 + 2.51T + 37T^{2} \) |
| 41 | \( 1 - 7.74T + 41T^{2} \) |
| 43 | \( 1 + 2.16T + 43T^{2} \) |
| 47 | \( 1 + 2.09T + 47T^{2} \) |
| 53 | \( 1 + 2.90T + 53T^{2} \) |
| 59 | \( 1 - 11.9T + 59T^{2} \) |
| 61 | \( 1 - 14.1T + 61T^{2} \) |
| 67 | \( 1 - 5.23T + 67T^{2} \) |
| 71 | \( 1 + 15.6T + 71T^{2} \) |
| 73 | \( 1 + 2.37T + 73T^{2} \) |
| 79 | \( 1 - 8.01T + 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 97 | \( 1 - 11.1T + 97T^{2} \) |
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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
−8.007294917933229428003830688567, −6.92790458415992255073818733522, −6.49117271278788941748316897200, −5.73315918069674280123992492284, −5.05018889989051012449841203145, −4.21188970948000948428580607337, −3.72117536233440341181546719176, −2.55141648145650303408222167113, −1.92644629222806934432442432333, 0,
1.92644629222806934432442432333, 2.55141648145650303408222167113, 3.72117536233440341181546719176, 4.21188970948000948428580607337, 5.05018889989051012449841203145, 5.73315918069674280123992492284, 6.49117271278788941748316897200, 6.92790458415992255073818733522, 8.007294917933229428003830688567
