| L(s) = 1 | − 2.25·3-s + 1.77·5-s + 3.34·7-s + 2.06·9-s + 0.330·11-s + 1.57·13-s − 4.00·15-s + 17-s + 0.491·19-s − 7.52·21-s − 5.78·23-s − 1.83·25-s + 2.10·27-s + 1.24·29-s + 7.02·31-s − 0.743·33-s + 5.95·35-s − 9.95·37-s − 3.53·39-s − 11.2·41-s − 7.93·43-s + 3.67·45-s + 3.56·47-s + 4.17·49-s − 2.25·51-s − 7.57·53-s + 0.588·55-s + ⋯ |
| L(s) = 1 | − 1.29·3-s + 0.796·5-s + 1.26·7-s + 0.687·9-s + 0.0996·11-s + 0.435·13-s − 1.03·15-s + 0.242·17-s + 0.112·19-s − 1.64·21-s − 1.20·23-s − 0.366·25-s + 0.405·27-s + 0.230·29-s + 1.26·31-s − 0.129·33-s + 1.00·35-s − 1.63·37-s − 0.566·39-s − 1.75·41-s − 1.20·43-s + 0.547·45-s + 0.520·47-s + 0.596·49-s − 0.315·51-s − 1.04·53-s + 0.0793·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 + 2.25T + 3T^{2} \) |
| 5 | \( 1 - 1.77T + 5T^{2} \) |
| 7 | \( 1 - 3.34T + 7T^{2} \) |
| 11 | \( 1 - 0.330T + 11T^{2} \) |
| 13 | \( 1 - 1.57T + 13T^{2} \) |
| 19 | \( 1 - 0.491T + 19T^{2} \) |
| 23 | \( 1 + 5.78T + 23T^{2} \) |
| 29 | \( 1 - 1.24T + 29T^{2} \) |
| 31 | \( 1 - 7.02T + 31T^{2} \) |
| 37 | \( 1 + 9.95T + 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 + 7.93T + 43T^{2} \) |
| 47 | \( 1 - 3.56T + 47T^{2} \) |
| 53 | \( 1 + 7.57T + 53T^{2} \) |
| 61 | \( 1 - 2.21T + 61T^{2} \) |
| 67 | \( 1 + 8.26T + 67T^{2} \) |
| 71 | \( 1 + 8.19T + 71T^{2} \) |
| 73 | \( 1 + 7.73T + 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 - 6.18T + 83T^{2} \) |
| 89 | \( 1 + 14.8T + 89T^{2} \) |
| 97 | \( 1 - 4.83T + 97T^{2} \) |
| 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
−7.36409429382813047964551121820, −6.56451246469140841220123255656, −6.02500489268521174581580209103, −5.41755740615020447810180716032, −4.89924143798031067652667931738, −4.20424658998526933131818853031, −3.08321057199531503014273829688, −1.81647721257277098583872416829, −1.38326510729422059299519059727, 0,
1.38326510729422059299519059727, 1.81647721257277098583872416829, 3.08321057199531503014273829688, 4.20424658998526933131818853031, 4.89924143798031067652667931738, 5.41755740615020447810180716032, 6.02500489268521174581580209103, 6.56451246469140841220123255656, 7.36409429382813047964551121820
