| L(s) = 1 | + 3.46·5-s + 24.2·7-s − 48·11-s − 41.5·13-s − 54·17-s − 4·19-s − 173.·23-s − 113·25-s + 162.·29-s − 58.8·31-s + 84·35-s + 325.·37-s − 294·41-s + 188·43-s + 505.·47-s + 245·49-s + 744.·53-s − 166.·55-s − 252·59-s + 90.0·61-s − 144·65-s + 628·67-s + 6.92·71-s + 1.00e3·73-s − 1.16e3·77-s + 1.34e3·79-s + 720·83-s + ⋯ |
| L(s) = 1 | + 0.309·5-s + 1.30·7-s − 1.31·11-s − 0.886·13-s − 0.770·17-s − 0.0482·19-s − 1.57·23-s − 0.904·25-s + 1.04·29-s − 0.341·31-s + 0.405·35-s + 1.44·37-s − 1.11·41-s + 0.666·43-s + 1.56·47-s + 0.714·49-s + 1.93·53-s − 0.407·55-s − 0.556·59-s + 0.189·61-s − 0.274·65-s + 1.14·67-s + 0.0115·71-s + 1.61·73-s − 1.72·77-s + 1.90·79-s + 0.952·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.023752685\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.023752685\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 3.46T + 125T^{2} \) |
| 7 | \( 1 - 24.2T + 343T^{2} \) |
| 11 | \( 1 + 48T + 1.33e3T^{2} \) |
| 13 | \( 1 + 41.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 54T + 4.91e3T^{2} \) |
| 19 | \( 1 + 4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 173.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 162.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 58.8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 325.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 294T + 6.89e4T^{2} \) |
| 43 | \( 1 - 188T + 7.95e4T^{2} \) |
| 47 | \( 1 - 505.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 744.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 252T + 2.05e5T^{2} \) |
| 61 | \( 1 - 90.0T + 2.26e5T^{2} \) |
| 67 | \( 1 - 628T + 3.00e5T^{2} \) |
| 71 | \( 1 - 6.92T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.00e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.34e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 720T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.48e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.82e3T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.438887117514914577726216529966, −7.926218367997248865946977698429, −7.33860569110996719232652059100, −6.21198395836220509596138147854, −5.38537309416874898751543371524, −4.76970075411715799693438002826, −3.95567177885665624344381629516, −2.38479635658726799049317114254, −2.12450256821341808860276300417, −0.61188199318278778601744147913,
0.61188199318278778601744147913, 2.12450256821341808860276300417, 2.38479635658726799049317114254, 3.95567177885665624344381629516, 4.76970075411715799693438002826, 5.38537309416874898751543371524, 6.21198395836220509596138147854, 7.33860569110996719232652059100, 7.926218367997248865946977698429, 8.438887117514914577726216529966
