| L(s) = 1 | − 1.41·2-s − 1.73·3-s + 2.00·4-s + 2.23i·5-s + 2.44·6-s + 12.6i·7-s − 2.82·8-s + 2.99·9-s − 3.16i·10-s + 6.64i·11-s − 3.46·12-s − 21.6·13-s − 17.8i·14-s − 3.87i·15-s + 4.00·16-s + 26.9i·17-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.500·4-s + 0.447i·5-s + 0.408·6-s + 1.80i·7-s − 0.353·8-s + 0.333·9-s − 0.316i·10-s + 0.604i·11-s − 0.288·12-s − 1.66·13-s − 1.27i·14-s − 0.258i·15-s + 0.250·16-s + 1.58i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.873 + 0.486i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.873 + 0.486i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4013312863\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4013312863\) |
| \(L(2)\) |
|
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 | \( 1 + 1.73T \) |
| 5 | \( 1 - 2.23iT \) |
| 23 | \( 1 + (11.1 + 20.0i)T \) |
| good | 7 | \( 1 - 12.6iT - 49T^{2} \) |
| 11 | \( 1 - 6.64iT - 121T^{2} \) |
| 13 | \( 1 + 21.6T + 169T^{2} \) |
| 17 | \( 1 - 26.9iT - 289T^{2} \) |
| 19 | \( 1 - 5.80iT - 361T^{2} \) |
| 29 | \( 1 - 50.7T + 841T^{2} \) |
| 31 | \( 1 - 0.790T + 961T^{2} \) |
| 37 | \( 1 - 11.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 0.0875T + 1.68e3T^{2} \) |
| 43 | \( 1 + 78.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 65.2T + 2.20e3T^{2} \) |
| 53 | \( 1 - 35.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 28.0T + 3.48e3T^{2} \) |
| 61 | \( 1 + 18.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 23.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 101.T + 5.04e3T^{2} \) |
| 73 | \( 1 + 111.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 84.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 137. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 46.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 20.6iT - 9.40e3T^{2} \) |
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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
−10.40205163647135305086706397737, −10.13387617624058939425636041953, −9.023449621481685389637659114097, −8.305340688016298680721053681031, −7.27455344371303754850173744768, −6.32002413827573327372822398734, −5.62664086498052947019209597194, −4.50491371407584596343169933524, −2.72552650869942562917882831124, −1.95253732986598042603499786904,
0.22203620755118839311185705363, 1.08097655645885586346965953995, 2.87066498264019962041884980819, 4.35092461521566134446309595249, 5.09584378200888147668783379832, 6.44914977822748347906079490794, 7.37520595600197072929836346935, 7.70432876387231353604285471193, 9.095374817619606491215411139770, 9.938037838026898664575157577182
