L(s) = 1 | − 1.95·2-s − 3-s + 1.82·4-s − 3.69·5-s + 1.95·6-s − 7-s + 0.333·8-s + 9-s + 7.23·10-s − 1.37·11-s − 1.82·12-s − 1.91·13-s + 1.95·14-s + 3.69·15-s − 4.31·16-s − 2.07·17-s − 1.95·18-s + 6.94·19-s − 6.76·20-s + 21-s + 2.68·22-s + 2.64·23-s − 0.333·24-s + 8.66·25-s + 3.75·26-s − 27-s − 1.82·28-s + ⋯ |
L(s) = 1 | − 1.38·2-s − 0.577·3-s + 0.914·4-s − 1.65·5-s + 0.798·6-s − 0.377·7-s + 0.117·8-s + 0.333·9-s + 2.28·10-s − 0.413·11-s − 0.528·12-s − 0.531·13-s + 0.523·14-s + 0.954·15-s − 1.07·16-s − 0.503·17-s − 0.461·18-s + 1.59·19-s − 1.51·20-s + 0.218·21-s + 0.572·22-s + 0.552·23-s − 0.0681·24-s + 1.73·25-s + 0.735·26-s − 0.192·27-s − 0.345·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1348386205\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1348386205\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 191 | \( 1 - T \) |
good | 2 | \( 1 + 1.95T + 2T^{2} \) |
| 5 | \( 1 + 3.69T + 5T^{2} \) |
| 11 | \( 1 + 1.37T + 11T^{2} \) |
| 13 | \( 1 + 1.91T + 13T^{2} \) |
| 17 | \( 1 + 2.07T + 17T^{2} \) |
| 19 | \( 1 - 6.94T + 19T^{2} \) |
| 23 | \( 1 - 2.64T + 23T^{2} \) |
| 29 | \( 1 + 4.43T + 29T^{2} \) |
| 31 | \( 1 + 5.32T + 31T^{2} \) |
| 37 | \( 1 - 1.70T + 37T^{2} \) |
| 41 | \( 1 + 9.08T + 41T^{2} \) |
| 43 | \( 1 + 3.79T + 43T^{2} \) |
| 47 | \( 1 + 0.337T + 47T^{2} \) |
| 53 | \( 1 + 4.43T + 53T^{2} \) |
| 59 | \( 1 + 7.11T + 59T^{2} \) |
| 61 | \( 1 + 4.27T + 61T^{2} \) |
| 67 | \( 1 - 7.63T + 67T^{2} \) |
| 71 | \( 1 + 7.55T + 71T^{2} \) |
| 73 | \( 1 + 0.633T + 73T^{2} \) |
| 79 | \( 1 - 12.8T + 79T^{2} \) |
| 83 | \( 1 + 11.2T + 83T^{2} \) |
| 89 | \( 1 - 4.72T + 89T^{2} \) |
| 97 | \( 1 + 15.6T + 97T^{2} \) |
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
−8.297632336724189359215126303445, −7.80160884444147528273672702575, −7.17555551523959177971842762170, −6.78382043245502675678489115006, −5.40830253269646129332888890588, −4.72254063542532921380585906996, −3.79213814768322180124721256145, −2.90750174412332452212521547299, −1.46072096517345118825255512748, −0.27969784270733591254006567375,
0.27969784270733591254006567375, 1.46072096517345118825255512748, 2.90750174412332452212521547299, 3.79213814768322180124721256145, 4.72254063542532921380585906996, 5.40830253269646129332888890588, 6.78382043245502675678489115006, 7.17555551523959177971842762170, 7.80160884444147528273672702575, 8.297632336724189359215126303445