| L(s) = 1 | − 2-s − 1.84·3-s + 4-s − 3.34·5-s + 1.84·6-s + 2.16·7-s − 8-s + 0.403·9-s + 3.34·10-s + 3.73·11-s − 1.84·12-s + 4.02·13-s − 2.16·14-s + 6.17·15-s + 16-s + 4.29·17-s − 0.403·18-s − 2.98·19-s − 3.34·20-s − 3.98·21-s − 3.73·22-s + 6.16·23-s + 1.84·24-s + 6.18·25-s − 4.02·26-s + 4.79·27-s + 2.16·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.06·3-s + 0.5·4-s − 1.49·5-s + 0.753·6-s + 0.816·7-s − 0.353·8-s + 0.134·9-s + 1.05·10-s + 1.12·11-s − 0.532·12-s + 1.11·13-s − 0.577·14-s + 1.59·15-s + 0.250·16-s + 1.04·17-s − 0.0950·18-s − 0.683·19-s − 0.747·20-s − 0.869·21-s − 0.796·22-s + 1.28·23-s + 0.376·24-s + 1.23·25-s − 0.789·26-s + 0.921·27-s + 0.408·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8452604299\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8452604299\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2003 | \( 1 - T \) |
| good | 3 | \( 1 + 1.84T + 3T^{2} \) |
| 5 | \( 1 + 3.34T + 5T^{2} \) |
| 7 | \( 1 - 2.16T + 7T^{2} \) |
| 11 | \( 1 - 3.73T + 11T^{2} \) |
| 13 | \( 1 - 4.02T + 13T^{2} \) |
| 17 | \( 1 - 4.29T + 17T^{2} \) |
| 19 | \( 1 + 2.98T + 19T^{2} \) |
| 23 | \( 1 - 6.16T + 23T^{2} \) |
| 29 | \( 1 - 7.30T + 29T^{2} \) |
| 31 | \( 1 - 3.71T + 31T^{2} \) |
| 37 | \( 1 + 10.1T + 37T^{2} \) |
| 41 | \( 1 - 3.69T + 41T^{2} \) |
| 43 | \( 1 + 2.64T + 43T^{2} \) |
| 47 | \( 1 + 4.03T + 47T^{2} \) |
| 53 | \( 1 - 4.27T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 + 4.32T + 61T^{2} \) |
| 67 | \( 1 - 11.9T + 67T^{2} \) |
| 71 | \( 1 + 9.20T + 71T^{2} \) |
| 73 | \( 1 - 1.57T + 73T^{2} \) |
| 79 | \( 1 + 6.40T + 79T^{2} \) |
| 83 | \( 1 - 12.2T + 83T^{2} \) |
| 89 | \( 1 + 12.0T + 89T^{2} \) |
| 97 | \( 1 + 7.79T + 97T^{2} \) |
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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
−8.537687798421570766508029785881, −7.85408075120752751902724595298, −6.87620359506444257283104594779, −6.52932158424125184200633313444, −5.50109193877907015287480214478, −4.71216719077519325516119213365, −3.89070863382448184981207285484, −3.07772667701774614048983189288, −1.37456681952361745697720345999, −0.71099101067337850289018821962,
0.71099101067337850289018821962, 1.37456681952361745697720345999, 3.07772667701774614048983189288, 3.89070863382448184981207285484, 4.71216719077519325516119213365, 5.50109193877907015287480214478, 6.52932158424125184200633313444, 6.87620359506444257283104594779, 7.85408075120752751902724595298, 8.537687798421570766508029785881
