| L(s) = 1 | + 1.41·2-s − 1.73i·3-s + 2.00·4-s + 2.23i·5-s − 2.44i·6-s + 2.82·8-s − 2.99·9-s + 3.16i·10-s + 11.5·11-s − 3.46i·12-s − 7.86i·13-s + 3.87·15-s + 4.00·16-s − 27.6i·17-s − 4.24·18-s + 31.4i·19-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577i·3-s + 0.500·4-s + 0.447i·5-s − 0.408i·6-s + 0.353·8-s − 0.333·9-s + 0.316i·10-s + 1.05·11-s − 0.288i·12-s − 0.604i·13-s + 0.258·15-s + 0.250·16-s − 1.62i·17-s − 0.235·18-s + 1.65i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.422041254\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.422041254\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 + 1.73iT \) |
| 5 | \( 1 - 2.23iT \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 11.5T + 121T^{2} \) |
| 13 | \( 1 + 7.86iT - 169T^{2} \) |
| 17 | \( 1 + 27.6iT - 289T^{2} \) |
| 19 | \( 1 - 31.4iT - 361T^{2} \) |
| 23 | \( 1 - 18.1T + 529T^{2} \) |
| 29 | \( 1 + 2.30T + 841T^{2} \) |
| 31 | \( 1 - 5.26iT - 961T^{2} \) |
| 37 | \( 1 - 1.98T + 1.36e3T^{2} \) |
| 41 | \( 1 + 22.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 49.8T + 1.84e3T^{2} \) |
| 47 | \( 1 + 76.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 57.0T + 2.80e3T^{2} \) |
| 59 | \( 1 - 70.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 67.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 98.1T + 4.48e3T^{2} \) |
| 71 | \( 1 + 34.2T + 5.04e3T^{2} \) |
| 73 | \( 1 + 19.4iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 90.4T + 6.24e3T^{2} \) |
| 83 | \( 1 - 133. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 11.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 72.3iT - 9.40e3T^{2} \) |
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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
−9.239320352336594756598166129581, −8.258680210701613119490413332376, −7.31290570386047877315726141675, −6.85538457328654725133303995321, −5.90313447095996253419069451850, −5.21585896984209224144446709675, −3.97884300906782904987766578845, −3.17717992360122619116494963846, −2.14817881326565466645172488689, −0.883169469323541860650306153301,
1.13920176287800875146265963750, 2.43416117319605631830497932528, 3.66961955264809084812135433299, 4.31973685227646271691846073946, 5.04667673392966742942381906897, 6.10866552843368840237487736546, 6.70214217306054636590356554008, 7.76777368839638856038461407354, 8.907695438796355939916228963129, 9.192067145600643020133126523097
