| L(s) = 1 | + (−1.41 + i)3-s + (1.00 − 2.82i)9-s − 2.82i·11-s − 5.65·17-s + 2·19-s + (1.41 + 5.00i)27-s + (2.82 + 4.00i)33-s + 11.3i·41-s + 10i·43-s − 7·49-s + (8.00 − 5.65i)51-s + (−2.82 + 2i)57-s − 14.1i·59-s + 14i·67-s + 2i·73-s + ⋯ |
| L(s) = 1 | + (−0.816 + 0.577i)3-s + (0.333 − 0.942i)9-s − 0.852i·11-s − 1.37·17-s + 0.458·19-s + (0.272 + 0.962i)27-s + (0.492 + 0.696i)33-s + 1.76i·41-s + 1.52i·43-s − 49-s + (1.12 − 0.792i)51-s + (−0.374 + 0.264i)57-s − 1.84i·59-s + 1.71i·67-s + 0.234i·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 - 0.472i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.881 - 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4216305659\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4216305659\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.41 - i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 2.82iT - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 5.65T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 11.3iT - 41T^{2} \) |
| 43 | \( 1 - 10iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 14.1iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 14iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 - 2.82T + 83T^{2} \) |
| 89 | \( 1 - 5.65iT - 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−9.424888659609537304758950988763, −8.607843011815027373766475013817, −7.80943905238900101324431945307, −6.61963242881428181161187029389, −6.29271407655236427610744979862, −5.28303009069981396384737105043, −4.61765614760641143620334831032, −3.73192359101095674866413692047, −2.76257739523020073045330005363, −1.20863639602178718507715692453,
0.17013275168272142208483974471, 1.64630922937879904292858501678, 2.47819439129769698588424574198, 3.94142737416113330815536991514, 4.78530692609946303248675764718, 5.49856286604428260017892730429, 6.41804476956842995249193085505, 7.06276705956025717116034359040, 7.62382553203900393763334923509, 8.635220474065456136030836545036
