| L(s) = 1 | − 3-s − 2.60·5-s − 7-s + 9-s + 3.69·11-s − 5.98·13-s + 2.60·15-s − 5.23·17-s + 3.69·19-s + 21-s − 23-s + 1.77·25-s − 27-s − 4.36·29-s + 4.00·31-s − 3.69·33-s + 2.60·35-s − 8.22·37-s + 5.98·39-s + 6.55·41-s − 8.60·43-s − 2.60·45-s − 2.68·47-s + 49-s + 5.23·51-s + 11.3·53-s − 9.62·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.16·5-s − 0.377·7-s + 0.333·9-s + 1.11·11-s − 1.65·13-s + 0.672·15-s − 1.26·17-s + 0.847·19-s + 0.218·21-s − 0.208·23-s + 0.355·25-s − 0.192·27-s − 0.811·29-s + 0.718·31-s − 0.643·33-s + 0.440·35-s − 1.35·37-s + 0.958·39-s + 1.02·41-s − 1.31·43-s − 0.388·45-s − 0.391·47-s + 0.142·49-s + 0.732·51-s + 1.55·53-s − 1.29·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6279536955\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6279536955\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + 2.60T + 5T^{2} \) |
| 11 | \( 1 - 3.69T + 11T^{2} \) |
| 13 | \( 1 + 5.98T + 13T^{2} \) |
| 17 | \( 1 + 5.23T + 17T^{2} \) |
| 19 | \( 1 - 3.69T + 19T^{2} \) |
| 29 | \( 1 + 4.36T + 29T^{2} \) |
| 31 | \( 1 - 4.00T + 31T^{2} \) |
| 37 | \( 1 + 8.22T + 37T^{2} \) |
| 41 | \( 1 - 6.55T + 41T^{2} \) |
| 43 | \( 1 + 8.60T + 43T^{2} \) |
| 47 | \( 1 + 2.68T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 + 14.6T + 59T^{2} \) |
| 61 | \( 1 - 4.08T + 61T^{2} \) |
| 67 | \( 1 - 3.79T + 67T^{2} \) |
| 71 | \( 1 + 1.77T + 71T^{2} \) |
| 73 | \( 1 - 1.52T + 73T^{2} \) |
| 79 | \( 1 - 1.97T + 79T^{2} \) |
| 83 | \( 1 + 7.08T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 - 1.66T + 97T^{2} \) |
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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
−8.450277075129797391498003327735, −7.53672728446021000146446468456, −7.05074900511319147866179496997, −6.45655390248208981935215677327, −5.40751085102089649110397640424, −4.60028411304398719458767929832, −4.01091649848217638750655751420, −3.12574135764051552719960592077, −1.92243401513003877793563418370, −0.45584450918207657264681706070,
0.45584450918207657264681706070, 1.92243401513003877793563418370, 3.12574135764051552719960592077, 4.01091649848217638750655751420, 4.60028411304398719458767929832, 5.40751085102089649110397640424, 6.45655390248208981935215677327, 7.05074900511319147866179496997, 7.53672728446021000146446468456, 8.450277075129797391498003327735
