| L(s) = 1 | − 2.92i·2-s − 4.55·4-s + 2.23i·5-s − 2.18·7-s + 1.63i·8-s + 6.54·10-s + 3.31i·11-s − 9.32·13-s + 6.38i·14-s − 13.4·16-s + 6.11i·17-s − 28.8·19-s − 10.1i·20-s + 9.70·22-s + 31.8i·23-s + ⋯ |
| L(s) = 1 | − 1.46i·2-s − 1.13·4-s + 0.447i·5-s − 0.312·7-s + 0.204i·8-s + 0.654·10-s + 0.301i·11-s − 0.717·13-s + 0.456i·14-s − 0.840·16-s + 0.359i·17-s − 1.51·19-s − 0.509i·20-s + 0.441·22-s + 1.38i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4073511756\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4073511756\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
| 11 | \( 1 - 3.31iT \) |
| good | 2 | \( 1 + 2.92iT - 4T^{2} \) |
| 7 | \( 1 + 2.18T + 49T^{2} \) |
| 13 | \( 1 + 9.32T + 169T^{2} \) |
| 17 | \( 1 - 6.11iT - 289T^{2} \) |
| 19 | \( 1 + 28.8T + 361T^{2} \) |
| 23 | \( 1 - 31.8iT - 529T^{2} \) |
| 29 | \( 1 - 16.1iT - 841T^{2} \) |
| 31 | \( 1 - 14.9T + 961T^{2} \) |
| 37 | \( 1 - 21.4T + 1.36e3T^{2} \) |
| 41 | \( 1 - 31.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 34.3T + 1.84e3T^{2} \) |
| 47 | \( 1 + 42.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 51.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 22.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 67.3T + 3.72e3T^{2} \) |
| 67 | \( 1 - 107.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 21.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 44.1T + 5.32e3T^{2} \) |
| 79 | \( 1 + 157.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 95.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 98.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 35.1T + 9.40e3T^{2} \) |
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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
−10.93763120242280047779756031317, −10.07299338473499555942037671138, −9.575872305410891878879709911456, −8.451819383064372948077983679501, −7.21661379905818267150057038120, −6.26544714496001023213058921564, −4.78919089835429499484954789298, −3.74970460079506258348712320153, −2.74447323580443396654158454698, −1.67831030169556038163096506446,
0.15458672077625544275430043598, 2.44876049145853732481190738960, 4.26682358540648943284948469038, 5.06623598713517445200461650619, 6.19175966262179152761493350142, 6.76242777684887722716266737635, 7.895790944957750858235381993076, 8.522224826919408205913572909743, 9.378305054473598144950253107536, 10.41766193013506027212378810952
