| L(s) = 1 | + 1.60·2-s − 3-s + 0.565·4-s + 2.27·5-s − 1.60·6-s + 7-s − 2.29·8-s + 9-s + 3.64·10-s + 2.53·11-s − 0.565·12-s + 1.60·14-s − 2.27·15-s − 4.81·16-s − 4.05·17-s + 1.60·18-s + 7.93·19-s + 1.28·20-s − 21-s + 4.06·22-s + 0.458·23-s + 2.29·24-s + 0.188·25-s − 27-s + 0.565·28-s + 6.01·29-s − 3.64·30-s + ⋯ |
| L(s) = 1 | + 1.13·2-s − 0.577·3-s + 0.282·4-s + 1.01·5-s − 0.653·6-s + 0.377·7-s − 0.812·8-s + 0.333·9-s + 1.15·10-s + 0.765·11-s − 0.163·12-s + 0.428·14-s − 0.588·15-s − 1.20·16-s − 0.983·17-s + 0.377·18-s + 1.82·19-s + 0.287·20-s − 0.218·21-s + 0.867·22-s + 0.0955·23-s + 0.469·24-s + 0.0377·25-s − 0.192·27-s + 0.106·28-s + 1.11·29-s − 0.666·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.406168076\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.406168076\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.60T + 2T^{2} \) |
| 5 | \( 1 - 2.27T + 5T^{2} \) |
| 11 | \( 1 - 2.53T + 11T^{2} \) |
| 17 | \( 1 + 4.05T + 17T^{2} \) |
| 19 | \( 1 - 7.93T + 19T^{2} \) |
| 23 | \( 1 - 0.458T + 23T^{2} \) |
| 29 | \( 1 - 6.01T + 29T^{2} \) |
| 31 | \( 1 + 0.340T + 31T^{2} \) |
| 37 | \( 1 - 4.74T + 37T^{2} \) |
| 41 | \( 1 - 8.88T + 41T^{2} \) |
| 43 | \( 1 + 2.14T + 43T^{2} \) |
| 47 | \( 1 + 7.16T + 47T^{2} \) |
| 53 | \( 1 - 3.30T + 53T^{2} \) |
| 59 | \( 1 - 4.30T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 - 4.25T + 71T^{2} \) |
| 73 | \( 1 - 1.59T + 73T^{2} \) |
| 79 | \( 1 + 9.42T + 79T^{2} \) |
| 83 | \( 1 - 1.87T + 83T^{2} \) |
| 89 | \( 1 - 17.9T + 89T^{2} \) |
| 97 | \( 1 - 18.9T + 97T^{2} \) |
| 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
−8.712472146685192982551121294833, −7.61181811869651768388170447224, −6.63565006094479940862964678381, −6.17601029416554820657640976888, −5.45244124136640057342312055670, −4.85652619577363879694171163624, −4.15628061116241423101150558246, −3.15804234469049938943099475296, −2.17142285782326617460524501414, −0.991284209189865882699352850025,
0.991284209189865882699352850025, 2.17142285782326617460524501414, 3.15804234469049938943099475296, 4.15628061116241423101150558246, 4.85652619577363879694171163624, 5.45244124136640057342312055670, 6.17601029416554820657640976888, 6.63565006094479940862964678381, 7.61181811869651768388170447224, 8.712472146685192982551121294833
