| L(s) = 1 | + 4.24i·5-s + 4·7-s + 16.9i·11-s + 8·13-s + 12.7i·17-s + 16·19-s − 16.9i·23-s + 7.00·25-s − 4.24i·29-s − 44·31-s + 16.9i·35-s − 34·37-s − 46.6i·41-s + 40·43-s − 84.8i·47-s + ⋯ |
| L(s) = 1 | + 0.848i·5-s + 0.571·7-s + 1.54i·11-s + 0.615·13-s + 0.748i·17-s + 0.842·19-s − 0.737i·23-s + 0.280·25-s − 0.146i·29-s − 1.41·31-s + 0.484i·35-s − 0.918·37-s − 1.13i·41-s + 0.930·43-s − 1.80i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.31967 + 0.683116i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.31967 + 0.683116i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 4.24iT - 25T^{2} \) |
| 7 | \( 1 - 4T + 49T^{2} \) |
| 11 | \( 1 - 16.9iT - 121T^{2} \) |
| 13 | \( 1 - 8T + 169T^{2} \) |
| 17 | \( 1 - 12.7iT - 289T^{2} \) |
| 19 | \( 1 - 16T + 361T^{2} \) |
| 23 | \( 1 + 16.9iT - 529T^{2} \) |
| 29 | \( 1 + 4.24iT - 841T^{2} \) |
| 31 | \( 1 + 44T + 961T^{2} \) |
| 37 | \( 1 + 34T + 1.36e3T^{2} \) |
| 41 | \( 1 + 46.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 40T + 1.84e3T^{2} \) |
| 47 | \( 1 + 84.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 38.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 33.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 50T + 3.72e3T^{2} \) |
| 67 | \( 1 + 8T + 4.48e3T^{2} \) |
| 71 | \( 1 + 50.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 16T + 5.32e3T^{2} \) |
| 79 | \( 1 - 76T + 6.24e3T^{2} \) |
| 83 | \( 1 - 118. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 12.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 176T + 9.40e3T^{2} \) |
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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
−12.95144612329030438916692540454, −11.99285281882888634036664730608, −10.87886488031190005676882756420, −10.14994928887121089115636409590, −8.859447563945669516645463318685, −7.56127038151087732071704044096, −6.70842282145024197873733413351, −5.21592030494207403613256592313, −3.77186660978585751289383881603, −2.01410154362396858144121923750,
1.11813309660071895897224950303, 3.35682349327906350185366723474, 4.94699627927993528046479845951, 5.92088429781282570169664953717, 7.56535336701896195563304015038, 8.602559935029808092750102300036, 9.366793536857037910290904970593, 10.95036025816715436989989699316, 11.55734411058502514096370369368, 12.78812196102597744355154839078
