L(s) = 1 | + (−1.5 + 0.866i)3-s + (1 + 1.73i)5-s − 11·7-s + (1.5 − 2.59i)9-s + 8·11-s + (−4.5 − 2.59i)13-s + (−3 − 1.73i)15-s + (13 + 22.5i)17-s − 19·19-s + (16.5 − 9.52i)21-s + (−16 + 27.7i)23-s + (10.5 − 18.1i)25-s + 5.19i·27-s + (−21 − 12.1i)29-s − 53.6i·31-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.288i)3-s + (0.200 + 0.346i)5-s − 1.57·7-s + (0.166 − 0.288i)9-s + 0.727·11-s + (−0.346 − 0.199i)13-s + (−0.200 − 0.115i)15-s + (0.764 + 1.32i)17-s − 19-s + (0.785 − 0.453i)21-s + (−0.695 + 1.20i)23-s + (0.419 − 0.727i)25-s + 0.192i·27-s + (−0.724 − 0.418i)29-s − 1.73i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.412 + 0.910i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.412 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7651286900\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7651286900\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.5 - 0.866i)T \) |
| 19 | \( 1 + 19T \) |
good | 5 | \( 1 + (-1 - 1.73i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + 11T + 49T^{2} \) |
| 11 | \( 1 - 8T + 121T^{2} \) |
| 13 | \( 1 + (4.5 + 2.59i)T + (84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + (-13 - 22.5i)T + (-144.5 + 250. i)T^{2} \) |
| 23 | \( 1 + (16 - 27.7i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (21 + 12.1i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + 53.6iT - 961T^{2} \) |
| 37 | \( 1 + 46.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (12 - 6.92i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-23.5 - 40.7i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-35 + 60.6i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (6 + 3.46i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-93 + 53.6i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (17.5 - 30.3i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (13.5 + 7.79i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + (-114 + 65.8i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (29.5 + 51.0i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-22.5 + 12.9i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 - 2T + 6.88e3T^{2} \) |
| 89 | \( 1 + (69 + 39.8i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (18 - 10.3i)T + (4.70e3 - 8.14e3i)T^{2} \) |
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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
−9.826193117897481048703634304477, −9.228323256304648841787987717975, −8.021596726805085442830975814789, −6.96876332075509595668309230635, −6.10539111789378810591762847352, −5.77549358539835775961093168791, −4.09572393445894342387781764718, −3.54911642293396019888450835512, −2.14426243533029911295299311466, −0.31981156867561010391259947647,
0.989809644471434917879001233345, 2.57038724343422322662046559313, 3.67711112334076759788917204580, 4.85701710422782470659853411768, 5.81781026168899803235473350410, 6.72538149481743279217537647881, 7.11647464851411437699259291730, 8.547901562177282876531661989983, 9.304943289029110078747822570104, 9.982506531401202940433795749999