L(s) = 1 | + (−0.866 + 0.5i)2-s + (−0.866 + 1.5i)3-s + (−1.5 + 2.59i)4-s − 1.73i·6-s + (−4.33 − 5.5i)7-s − 7i·8-s + (−1.5 − 2.59i)9-s + (−2 + 3.46i)11-s + (−2.59 − 4.5i)12-s + 17.3·13-s + (6.5 + 2.59i)14-s + (−2.5 − 4.33i)16-s + (4.33 − 7.5i)17-s + (2.59 + 1.5i)18-s + (−6 + 3.46i)19-s + ⋯ |
L(s) = 1 | + (−0.433 + 0.250i)2-s + (−0.288 + 0.5i)3-s + (−0.375 + 0.649i)4-s − 0.288i·6-s + (−0.618 − 0.785i)7-s − 0.875i·8-s + (−0.166 − 0.288i)9-s + (−0.181 + 0.314i)11-s + (−0.216 − 0.375i)12-s + 1.33·13-s + (0.464 + 0.185i)14-s + (−0.156 − 0.270i)16-s + (0.254 − 0.441i)17-s + (0.144 + 0.0833i)18-s + (−0.315 + 0.182i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.657 - 0.753i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.657 - 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9837839286\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9837839286\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.866 - 1.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (4.33 + 5.5i)T \) |
good | 2 | \( 1 + (0.866 - 0.5i)T + (2 - 3.46i)T^{2} \) |
| 11 | \( 1 + (2 - 3.46i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 - 17.3T + 169T^{2} \) |
| 17 | \( 1 + (-4.33 + 7.5i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (6 - 3.46i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-26.8 + 15.5i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 10T + 841T^{2} \) |
| 31 | \( 1 + (-25.5 - 14.7i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (43.3 - 25i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 - 53.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 34iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (21.6 + 37.5i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-43.3 - 25i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-48 - 27.7i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-21 + 12.1i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-43.3 - 25i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 97T + 5.04e3T^{2} \) |
| 73 | \( 1 + (24.2 - 42i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (3.5 + 6.06i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 - 152.T + 6.88e3T^{2} \) |
| 89 | \( 1 + (-136.5 + 78.8i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 112.T + 9.40e3T^{2} \) |
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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
−10.52141324793149680948422364228, −9.918782499603268584917969965386, −8.935890820324725397945130549925, −8.263834840485667593589448909875, −7.08943043252603179566222574219, −6.45166708284930817691737849279, −5.00084749813635171636286995034, −3.95931627557223396837567699814, −3.16566172996118700393786658903, −0.77798658495052976665593420504,
0.78258101752790498503656767067, 2.09706488989570714040400725556, 3.54664338165797659094602145812, 5.18612536046706502648949787513, 5.89217234514550403614343964077, 6.69083095538585588762847088957, 8.129425429424796761204217207840, 8.831132077072704407874201711090, 9.556731347976701766213895796279, 10.66446685898210091966430687992