| L(s) = 1 | − 1.83·2-s + 3-s + 1.36·4-s − 1.54·5-s − 1.83·6-s + 0.834·7-s + 1.16·8-s + 9-s + 2.83·10-s + 1.36·12-s + 13-s − 1.53·14-s − 1.54·15-s − 4.86·16-s − 3.96·17-s − 1.83·18-s − 1.49·19-s − 2.11·20-s + 0.834·21-s + 1.75·23-s + 1.16·24-s − 2.61·25-s − 1.83·26-s + 27-s + 1.14·28-s + 8.51·29-s + 2.83·30-s + ⋯ |
| L(s) = 1 | − 1.29·2-s + 0.577·3-s + 0.683·4-s − 0.690·5-s − 0.749·6-s + 0.315·7-s + 0.411·8-s + 0.333·9-s + 0.896·10-s + 0.394·12-s + 0.277·13-s − 0.409·14-s − 0.398·15-s − 1.21·16-s − 0.960·17-s − 0.432·18-s − 0.343·19-s − 0.471·20-s + 0.182·21-s + 0.366·23-s + 0.237·24-s − 0.522·25-s − 0.359·26-s + 0.192·27-s + 0.215·28-s + 1.58·29-s + 0.517·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4719 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4719 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 1.83T + 2T^{2} \) |
| 5 | \( 1 + 1.54T + 5T^{2} \) |
| 7 | \( 1 - 0.834T + 7T^{2} \) |
| 17 | \( 1 + 3.96T + 17T^{2} \) |
| 19 | \( 1 + 1.49T + 19T^{2} \) |
| 23 | \( 1 - 1.75T + 23T^{2} \) |
| 29 | \( 1 - 8.51T + 29T^{2} \) |
| 31 | \( 1 + 1.57T + 31T^{2} \) |
| 37 | \( 1 + 5.52T + 37T^{2} \) |
| 41 | \( 1 - 8.87T + 41T^{2} \) |
| 43 | \( 1 - 3.09T + 43T^{2} \) |
| 47 | \( 1 - 2.22T + 47T^{2} \) |
| 53 | \( 1 - 0.498T + 53T^{2} \) |
| 59 | \( 1 + 5.84T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 + 8.43T + 67T^{2} \) |
| 71 | \( 1 + 4.72T + 71T^{2} \) |
| 73 | \( 1 - 1.64T + 73T^{2} \) |
| 79 | \( 1 + 2.37T + 79T^{2} \) |
| 83 | \( 1 + 11.2T + 83T^{2} \) |
| 89 | \( 1 + 6.16T + 89T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.119880269537799910839194060066, −7.48507552272110263653730380463, −6.90275932372793686168564686005, −5.99840615078201538725842592525, −4.62891384125519218289144506915, −4.29360371421399297650208978356, −3.14538891062114636200731164416, −2.16496188133312366704469134833, −1.22269456485873716965983188067, 0,
1.22269456485873716965983188067, 2.16496188133312366704469134833, 3.14538891062114636200731164416, 4.29360371421399297650208978356, 4.62891384125519218289144506915, 5.99840615078201538725842592525, 6.90275932372793686168564686005, 7.48507552272110263653730380463, 8.119880269537799910839194060066
