| L(s) = 1 | + (−0.707 + 1.58i)3-s + 2i·5-s + (−2.00 − 2.23i)9-s + 1.41·11-s + 4.47·13-s + (−3.16 − 1.41i)15-s + 4.47i·17-s + 3.16i·19-s − 6.32·23-s + 25-s + (4.94 − 1.58i)27-s + 6i·29-s + 8.48i·31-s + (−1.00 + 2.23i)33-s + 4.47·37-s + ⋯ |
| L(s) = 1 | + (−0.408 + 0.912i)3-s + 0.894i·5-s + (−0.666 − 0.745i)9-s + 0.426·11-s + 1.24·13-s + (−0.816 − 0.365i)15-s + 1.08i·17-s + 0.725i·19-s − 1.31·23-s + 0.200·25-s + (0.952 − 0.304i)27-s + 1.11i·29-s + 1.52i·31-s + (−0.174 + 0.389i)33-s + 0.735·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 - 0.408i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.912 - 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.202973638\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.202973638\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.707 - 1.58i)T \) |
| good | 5 | \( 1 - 2iT - 5T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 - 4.47iT - 17T^{2} \) |
| 19 | \( 1 - 3.16iT - 19T^{2} \) |
| 23 | \( 1 + 6.32T + 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 - 8.48iT - 31T^{2} \) |
| 37 | \( 1 - 4.47T + 37T^{2} \) |
| 41 | \( 1 + 8.94iT - 41T^{2} \) |
| 43 | \( 1 + 3.16iT - 43T^{2} \) |
| 47 | \( 1 + 12.6T + 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 9.89T + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 + 3.16iT - 67T^{2} \) |
| 71 | \( 1 + 6.32T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + 2.82iT - 79T^{2} \) |
| 83 | \( 1 - 12.7T + 83T^{2} \) |
| 89 | \( 1 + 13.4iT - 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−10.04380299029656840805495627780, −8.961484513799796379183488453442, −8.455898313589027361102200267331, −7.30129277635750805372341564542, −6.15884960325266073597116905186, −6.04863012685910322422748567271, −4.71792727020004215665794048136, −3.71182992792332249649845163031, −3.25563549620173706072489051382, −1.59317310665977400438892138534,
0.52459963003954435971164271997, 1.54699639405070939864142387978, 2.77331550777408355047176255431, 4.18725322696641745156689832165, 4.99161515076286378908344026126, 6.06525941400834258018100441405, 6.44243769006208918673757764560, 7.69154493955441343564846749317, 8.136567265868629154864266009373, 9.061750033627422029103087656943
