| L(s) = 1 | + 2.23·2-s + 1.00·4-s + (2 − 4.58i)5-s − 11.1·7-s − 6.70·8-s + (4.47 − 10.2i)10-s + (4 + 10.2i)11-s − 8.94·13-s − 25.0·14-s − 19·16-s − 15.6·17-s − 10.2i·19-s + (2.00 − 4.58i)20-s + (8.94 + 22.9i)22-s + 27.4i·23-s + ⋯ |
| L(s) = 1 | + 1.11·2-s + 0.250·4-s + (0.400 − 0.916i)5-s − 1.59·7-s − 0.838·8-s + (0.447 − 1.02i)10-s + (0.363 + 0.931i)11-s − 0.688·13-s − 1.78·14-s − 1.18·16-s − 0.920·17-s − 0.539i·19-s + (0.100 − 0.229i)20-s + (0.406 + 1.04i)22-s + 1.19i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0393i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 - 0.0393i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3156562581\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3156562581\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-2 + 4.58i)T \) |
| 11 | \( 1 + (-4 - 10.2i)T \) |
| good | 2 | \( 1 - 2.23T + 4T^{2} \) |
| 7 | \( 1 + 11.1T + 49T^{2} \) |
| 13 | \( 1 + 8.94T + 169T^{2} \) |
| 17 | \( 1 + 15.6T + 289T^{2} \) |
| 19 | \( 1 + 10.2iT - 361T^{2} \) |
| 23 | \( 1 - 27.4iT - 529T^{2} \) |
| 29 | \( 1 + 10.2iT - 841T^{2} \) |
| 31 | \( 1 + 3T + 961T^{2} \) |
| 37 | \( 1 + 4.58iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 20.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 22.3T + 1.84e3T^{2} \) |
| 47 | \( 1 + 64.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 4.58iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 18T + 3.48e3T^{2} \) |
| 61 | \( 1 + 71.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 27.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 27T + 5.04e3T^{2} \) |
| 73 | \( 1 + 58.1T + 5.32e3T^{2} \) |
| 79 | \( 1 - 61.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 71.5T + 6.88e3T^{2} \) |
| 89 | \( 1 + 37T + 7.92e3T^{2} \) |
| 97 | \( 1 + 119. iT - 9.40e3T^{2} \) |
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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
−9.998152419811394938667759793491, −9.464201952526860652728242139876, −8.800951480946571216779074038387, −7.15675427137983809821372022363, −6.35947222441135673097491735291, −5.39607075478935583626232305218, −4.52633654737807756156526948189, −3.57178762590321475258741924921, −2.27365536415923108108243432451, −0.079968372685928295320777055885,
2.65685572249216599889185150263, 3.29226995427121400292943350867, 4.35010892858655956889383423495, 5.78404156615581566052892413825, 6.34136429822479655359214554957, 7.02573382524594838598421450333, 8.675301252705772907589129114999, 9.529699156918934882494503493907, 10.32087279168000475341521183627, 11.32303423503577293981850725832
