| L(s) = 1 | + 2-s + 2.77·3-s + 4-s + 2.77·6-s − 4.69·7-s + 8-s + 4.71·9-s + 6.40·11-s + 2.77·12-s + 1.06·13-s − 4.69·14-s + 16-s + 1.91·17-s + 4.71·18-s − 19-s − 13.0·21-s + 6.40·22-s + 1.79·23-s + 2.77·24-s + 1.06·26-s + 4.75·27-s − 4.69·28-s + 2.93·29-s − 5.55·31-s + 32-s + 17.7·33-s + 1.91·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.60·3-s + 0.5·4-s + 1.13·6-s − 1.77·7-s + 0.353·8-s + 1.57·9-s + 1.93·11-s + 0.801·12-s + 0.295·13-s − 1.25·14-s + 0.250·16-s + 0.465·17-s + 1.11·18-s − 0.229·19-s − 2.84·21-s + 1.36·22-s + 0.374·23-s + 0.566·24-s + 0.208·26-s + 0.915·27-s − 0.887·28-s + 0.545·29-s − 0.997·31-s + 0.176·32-s + 3.09·33-s + 0.328·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.842969743\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.842969743\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 2.77T + 3T^{2} \) |
| 7 | \( 1 + 4.69T + 7T^{2} \) |
| 11 | \( 1 - 6.40T + 11T^{2} \) |
| 13 | \( 1 - 1.06T + 13T^{2} \) |
| 17 | \( 1 - 1.91T + 17T^{2} \) |
| 23 | \( 1 - 1.79T + 23T^{2} \) |
| 29 | \( 1 - 2.93T + 29T^{2} \) |
| 31 | \( 1 + 5.55T + 31T^{2} \) |
| 37 | \( 1 + 11.4T + 37T^{2} \) |
| 41 | \( 1 + 1.14T + 41T^{2} \) |
| 43 | \( 1 + 3.55T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 + 8.69T + 53T^{2} \) |
| 59 | \( 1 + 5.63T + 59T^{2} \) |
| 61 | \( 1 + 3.39T + 61T^{2} \) |
| 67 | \( 1 + 8.82T + 67T^{2} \) |
| 71 | \( 1 + 1.42T + 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 + 1.96T + 79T^{2} \) |
| 83 | \( 1 - 16.2T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 14.9T + 97T^{2} \) |
| 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
−9.762641523858283954421643493673, −9.163397354863053682121633926093, −8.626159814920393922514024077340, −7.28025840620011693446166694910, −6.73853777901954683676849324847, −5.89772105921229902687536822883, −4.23652763277160090553895259888, −3.49589122725442479360417953773, −3.05165975502807886761171049816, −1.63930587319736415565967290568,
1.63930587319736415565967290568, 3.05165975502807886761171049816, 3.49589122725442479360417953773, 4.23652763277160090553895259888, 5.89772105921229902687536822883, 6.73853777901954683676849324847, 7.28025840620011693446166694910, 8.626159814920393922514024077340, 9.163397354863053682121633926093, 9.762641523858283954421643493673
