| L(s) = 1 | − 1.70·3-s − 1.92·5-s + 0.214·7-s − 0.0878·9-s + 13-s + 3.28·15-s + 6.86·17-s + 3.87·19-s − 0.366·21-s + 3.40·23-s − 1.29·25-s + 5.26·27-s + 0.260·29-s − 1.36·31-s − 0.413·35-s − 1.72·37-s − 1.70·39-s + 1.34·41-s − 1.40·43-s + 0.169·45-s − 12.0·47-s − 6.95·49-s − 11.7·51-s − 6.85·53-s − 6.61·57-s − 3.71·59-s + 10.1·61-s + ⋯ |
| L(s) = 1 | − 0.985·3-s − 0.860·5-s + 0.0812·7-s − 0.0292·9-s + 0.277·13-s + 0.848·15-s + 1.66·17-s + 0.889·19-s − 0.0800·21-s + 0.709·23-s − 0.258·25-s + 1.01·27-s + 0.0482·29-s − 0.244·31-s − 0.0699·35-s − 0.283·37-s − 0.273·39-s + 0.210·41-s − 0.214·43-s + 0.0251·45-s − 1.75·47-s − 0.993·49-s − 1.64·51-s − 0.941·53-s − 0.876·57-s − 0.483·59-s + 1.29·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9710957558\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9710957558\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 1.70T + 3T^{2} \) |
| 5 | \( 1 + 1.92T + 5T^{2} \) |
| 7 | \( 1 - 0.214T + 7T^{2} \) |
| 17 | \( 1 - 6.86T + 17T^{2} \) |
| 19 | \( 1 - 3.87T + 19T^{2} \) |
| 23 | \( 1 - 3.40T + 23T^{2} \) |
| 29 | \( 1 - 0.260T + 29T^{2} \) |
| 31 | \( 1 + 1.36T + 31T^{2} \) |
| 37 | \( 1 + 1.72T + 37T^{2} \) |
| 41 | \( 1 - 1.34T + 41T^{2} \) |
| 43 | \( 1 + 1.40T + 43T^{2} \) |
| 47 | \( 1 + 12.0T + 47T^{2} \) |
| 53 | \( 1 + 6.85T + 53T^{2} \) |
| 59 | \( 1 + 3.71T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 + 8.18T + 67T^{2} \) |
| 71 | \( 1 - 9.30T + 71T^{2} \) |
| 73 | \( 1 + 5.70T + 73T^{2} \) |
| 79 | \( 1 + 3.47T + 79T^{2} \) |
| 83 | \( 1 - 9.97T + 83T^{2} \) |
| 89 | \( 1 + 6.53T + 89T^{2} \) |
| 97 | \( 1 - 13.5T + 97T^{2} \) |
| 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
−7.957197882409286820828371283142, −7.36688263701311035693037950038, −6.58357147349396571821719109484, −5.82319736153447772651109925013, −5.22551530946240670586253632585, −4.61630720566719747607696553052, −3.49774623331168985638456584303, −3.09771493840970675718405891794, −1.52037347269574472110421767917, −0.57078889996361315587179071399,
0.57078889996361315587179071399, 1.52037347269574472110421767917, 3.09771493840970675718405891794, 3.49774623331168985638456584303, 4.61630720566719747607696553052, 5.22551530946240670586253632585, 5.82319736153447772651109925013, 6.58357147349396571821719109484, 7.36688263701311035693037950038, 7.957197882409286820828371283142
