| L(s) = 1 | − 4·11-s + 2·13-s + 4·17-s + 4·23-s − 5·25-s − 4·29-s − 8·31-s + 2·37-s − 4·41-s + 8·43-s − 8·47-s + 4·53-s + 8·59-s − 2·61-s + 8·67-s + 12·71-s − 6·73-s + 8·79-s + 16·83-s − 12·89-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.20·11-s + 0.554·13-s + 0.970·17-s + 0.834·23-s − 25-s − 0.742·29-s − 1.43·31-s + 0.328·37-s − 0.624·41-s + 1.21·43-s − 1.16·47-s + 0.549·53-s + 1.04·59-s − 0.256·61-s + 0.977·67-s + 1.42·71-s − 0.702·73-s + 0.900·79-s + 1.75·83-s − 1.27·89-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−15.47001550768454, −14.93827738749618, −14.52685456470823, −13.76041452344528, −13.38162906875889, −12.77199132048785, −12.48455961159363, −11.60651265255363, −11.18632674059646, −10.65348513033923, −10.12821261557216, −9.466937800150938, −9.064040205841096, −8.108605351042311, −7.942184698377683, −7.242983554369537, −6.648359024682677, −5.738361525240832, −5.461748484866907, −4.864571076330555, −3.845351414194277, −3.493405632311776, −2.617990288877037, −1.941450814930834, −1.013552328774535, 0,
1.013552328774535, 1.941450814930834, 2.617990288877037, 3.493405632311776, 3.845351414194277, 4.864571076330555, 5.461748484866907, 5.738361525240832, 6.648359024682677, 7.242983554369537, 7.942184698377683, 8.108605351042311, 9.064040205841096, 9.466937800150938, 10.12821261557216, 10.65348513033923, 11.18632674059646, 11.60651265255363, 12.48455961159363, 12.77199132048785, 13.38162906875889, 13.76041452344528, 14.52685456470823, 14.93827738749618, 15.47001550768454
