| L(s) = 1 | + i·5-s + 1.27i·7-s − 5.43i·11-s − 0.166·13-s + (−2.27 + 3.43i)17-s − 2.54·19-s + 3.27i·23-s − 25-s − 6.54i·29-s + 5.98i·31-s − 1.27·35-s + 4i·37-s + 8.54i·41-s + 2.71·43-s − 10.7·47-s + ⋯ |
| L(s) = 1 | + 0.447i·5-s + 0.481i·7-s − 1.64i·11-s − 0.0462·13-s + (−0.551 + 0.834i)17-s − 0.584·19-s + 0.682i·23-s − 0.200·25-s − 1.21i·29-s + 1.07i·31-s − 0.215·35-s + 0.657i·37-s + 1.33i·41-s + 0.413·43-s − 1.56·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.834 - 0.551i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.834 - 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6711392741\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6711392741\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 17 | \( 1 + (2.27 - 3.43i)T \) |
| good | 7 | \( 1 - 1.27iT - 7T^{2} \) |
| 11 | \( 1 + 5.43iT - 11T^{2} \) |
| 13 | \( 1 + 0.166T + 13T^{2} \) |
| 19 | \( 1 + 2.54T + 19T^{2} \) |
| 23 | \( 1 - 3.27iT - 23T^{2} \) |
| 29 | \( 1 + 6.54iT - 29T^{2} \) |
| 31 | \( 1 - 5.98iT - 31T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 - 8.54iT - 41T^{2} \) |
| 43 | \( 1 - 2.71T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 + 12.8T + 53T^{2} \) |
| 59 | \( 1 - 6.21T + 59T^{2} \) |
| 61 | \( 1 - 7.42iT - 61T^{2} \) |
| 67 | \( 1 + 6.37T + 67T^{2} \) |
| 71 | \( 1 + 2.89iT - 71T^{2} \) |
| 73 | \( 1 - 12.5iT - 73T^{2} \) |
| 79 | \( 1 + 3.77iT - 79T^{2} \) |
| 83 | \( 1 + 13.5T + 83T^{2} \) |
| 89 | \( 1 - 5.83T + 89T^{2} \) |
| 97 | \( 1 - 1.45iT - 97T^{2} \) |
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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
−8.798006442283898407677556890032, −8.386482687592093984742465971168, −7.64878825578978641539816551920, −6.45952984123086398037566976989, −6.17746549780647183245678053073, −5.31511307720227412155032238706, −4.26118286706903552549002999311, −3.35117211423754297857808583650, −2.63246333347361489233532819958, −1.40680726499751443631965724090,
0.20288957926920770801515475396, 1.67452580691878678283372438502, 2.51510301520131962114895912653, 3.82263799997062746754901957996, 4.58898501778688126653056203736, 5.07708232868816742492189945694, 6.23857465340909506782911257315, 7.05517520576750893263638682508, 7.49895748871403190826022778451, 8.445110063312045677850390764464
