| L(s) = 1 | + 2.13·2-s − 3-s + 2.54·4-s − 0.258·5-s − 2.13·6-s + 1.82·7-s + 1.16·8-s + 9-s − 0.551·10-s − 1.40·11-s − 2.54·12-s − 13-s + 3.90·14-s + 0.258·15-s − 2.60·16-s − 4.77·17-s + 2.13·18-s − 0.399·19-s − 0.659·20-s − 1.82·21-s − 3.00·22-s − 2.95·23-s − 1.16·24-s − 4.93·25-s − 2.13·26-s − 27-s + 4.66·28-s + ⋯ |
| L(s) = 1 | + 1.50·2-s − 0.577·3-s + 1.27·4-s − 0.115·5-s − 0.870·6-s + 0.691·7-s + 0.412·8-s + 0.333·9-s − 0.174·10-s − 0.424·11-s − 0.735·12-s − 0.277·13-s + 1.04·14-s + 0.0668·15-s − 0.651·16-s − 1.15·17-s + 0.502·18-s − 0.0917·19-s − 0.147·20-s − 0.399·21-s − 0.639·22-s − 0.616·23-s − 0.238·24-s − 0.986·25-s − 0.418·26-s − 0.192·27-s + 0.880·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 - 2.13T + 2T^{2} \) |
| 5 | \( 1 + 0.258T + 5T^{2} \) |
| 7 | \( 1 - 1.82T + 7T^{2} \) |
| 11 | \( 1 + 1.40T + 11T^{2} \) |
| 17 | \( 1 + 4.77T + 17T^{2} \) |
| 19 | \( 1 + 0.399T + 19T^{2} \) |
| 23 | \( 1 + 2.95T + 23T^{2} \) |
| 29 | \( 1 - 3.35T + 29T^{2} \) |
| 31 | \( 1 + 8.38T + 31T^{2} \) |
| 37 | \( 1 + 4.14T + 37T^{2} \) |
| 41 | \( 1 - 4.99T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 - 3.24T + 47T^{2} \) |
| 53 | \( 1 + 5.67T + 53T^{2} \) |
| 59 | \( 1 - 6.34T + 59T^{2} \) |
| 61 | \( 1 + 9.40T + 61T^{2} \) |
| 67 | \( 1 + 3.90T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 + 4.15T + 73T^{2} \) |
| 79 | \( 1 - 12.5T + 79T^{2} \) |
| 83 | \( 1 + 13.1T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 - 18.4T + 97T^{2} \) |
| 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
−7.75222819248267403497744396129, −7.18825253440255226427654066536, −6.24332038748359258873028928557, −5.74800093011871443282586083209, −4.96946216717536994823207366645, −4.40347018644374298809436513779, −3.76337788371268890747847625405, −2.60116581504971082553513045302, −1.81196913546834979480727903271, 0,
1.81196913546834979480727903271, 2.60116581504971082553513045302, 3.76337788371268890747847625405, 4.40347018644374298809436513779, 4.96946216717536994823207366645, 5.74800093011871443282586083209, 6.24332038748359258873028928557, 7.18825253440255226427654066536, 7.75222819248267403497744396129
