L(s) = 1 | + 5.52·3-s − 2.23i·5-s + 3.78i·7-s + 21.5·9-s − 2.97i·11-s − 0.383·13-s − 12.3i·15-s − 7.29i·17-s − 21.5i·19-s + 20.8i·21-s + (13.1 − 18.8i)23-s − 5.00·25-s + 69.0·27-s − 15.0·29-s + 11.1·31-s + ⋯ |
L(s) = 1 | + 1.84·3-s − 0.447i·5-s + 0.540i·7-s + 2.38·9-s − 0.270i·11-s − 0.0295·13-s − 0.823i·15-s − 0.428i·17-s − 1.13i·19-s + 0.994i·21-s + (0.573 − 0.819i)23-s − 0.200·25-s + 2.55·27-s − 0.519·29-s + 0.358·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 + 0.573i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.819 + 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(4.348171773\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.348171773\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 23 | \( 1 + (-13.1 + 18.8i)T \) |
good | 3 | \( 1 - 5.52T + 9T^{2} \) |
| 7 | \( 1 - 3.78iT - 49T^{2} \) |
| 11 | \( 1 + 2.97iT - 121T^{2} \) |
| 13 | \( 1 + 0.383T + 169T^{2} \) |
| 17 | \( 1 + 7.29iT - 289T^{2} \) |
| 19 | \( 1 + 21.5iT - 361T^{2} \) |
| 29 | \( 1 + 15.0T + 841T^{2} \) |
| 31 | \( 1 - 11.1T + 961T^{2} \) |
| 37 | \( 1 + 39.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 53.5T + 1.68e3T^{2} \) |
| 43 | \( 1 + 33.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 10.3T + 2.20e3T^{2} \) |
| 53 | \( 1 - 65.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 10.7T + 3.48e3T^{2} \) |
| 61 | \( 1 - 55.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 102. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 64.5T + 5.04e3T^{2} \) |
| 73 | \( 1 - 107.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 13.1iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 123. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 71.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 81.7iT - 9.40e3T^{2} \) |
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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
−9.048279765096053272140693159760, −8.435350571723132785418845106167, −7.54438755940599106639811616266, −6.97341330359398360985613349900, −5.71493419980337384768846751967, −4.63741036246511882782739106264, −3.89249900580917983928632433800, −2.76447955576158612730251259764, −2.33345912732435484827119338677, −0.943413906198053127241918030875,
1.35872479371920922527173219395, 2.26347371842522641187664818930, 3.30490051261860158498225737902, 3.79934079444435066777260830588, 4.76024246290362915829028181665, 6.15881049255274779542918390117, 7.10020575604916173223651453320, 7.74057989296443135628542133027, 8.215566444668180843057428851055, 9.174110309876906000671724552887