| L(s) = 1 | − 4·7-s − 4·13-s + 8·19-s − 4·23-s − 6·29-s − 8·31-s + 4·37-s − 6·41-s − 4·43-s + 4·47-s + 9·49-s − 12·53-s + 6·61-s − 12·67-s + 16·71-s − 8·79-s − 12·83-s + 10·89-s + 16·91-s + 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 1.10·13-s + 1.83·19-s − 0.834·23-s − 1.11·29-s − 1.43·31-s + 0.657·37-s − 0.937·41-s − 0.609·43-s + 0.583·47-s + 9/7·49-s − 1.64·53-s + 0.768·61-s − 1.46·67-s + 1.89·71-s − 0.900·79-s − 1.31·83-s + 1.05·89-s + 1.67·91-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8092674744\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8092674744\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−16.08269000886958, −15.68379251201532, −15.07447502699330, −14.31930069113060, −13.98098885814289, −13.08162202242730, −12.91316352939579, −12.18295545827683, −11.70189707621302, −11.07399568047675, −10.13369219763784, −9.818147068102598, −9.391561602214619, −8.806834648523250, −7.693712555721843, −7.444091017121557, −6.760634872885833, −6.042831217845396, −5.460496371088887, −4.814591396990432, −3.745738135564742, −3.346379311407920, −2.591217227405881, −1.671323366428086, −0.3811642083682741,
0.3811642083682741, 1.671323366428086, 2.591217227405881, 3.346379311407920, 3.745738135564742, 4.814591396990432, 5.460496371088887, 6.042831217845396, 6.760634872885833, 7.444091017121557, 7.693712555721843, 8.806834648523250, 9.391561602214619, 9.818147068102598, 10.13369219763784, 11.07399568047675, 11.70189707621302, 12.18295545827683, 12.91316352939579, 13.08162202242730, 13.98098885814289, 14.31930069113060, 15.07447502699330, 15.68379251201532, 16.08269000886958
