L(s) = 1 | − 1.41i·2-s − 2.00·4-s + (4.91 − 0.905i)5-s + (1.91 − 6.73i)7-s + 2.82i·8-s + (−1.28 − 6.95i)10-s − 17.5·11-s − 4.83·13-s + (−9.52 − 2.70i)14-s + 4.00·16-s − 18.0·17-s − 9.13i·19-s + (−9.83 + 1.81i)20-s + 24.7i·22-s + 3.72i·23-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.500·4-s + (0.983 − 0.181i)5-s + (0.273 − 0.961i)7-s + 0.353i·8-s + (−0.128 − 0.695i)10-s − 1.59·11-s − 0.371·13-s + (−0.680 − 0.193i)14-s + 0.250·16-s − 1.06·17-s − 0.480i·19-s + (−0.491 + 0.0905i)20-s + 1.12i·22-s + 0.161i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0946i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.995 - 0.0946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.039336543\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.039336543\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-4.91 + 0.905i)T \) |
| 7 | \( 1 + (-1.91 + 6.73i)T \) |
good | 11 | \( 1 + 17.5T + 121T^{2} \) |
| 13 | \( 1 + 4.83T + 169T^{2} \) |
| 17 | \( 1 + 18.0T + 289T^{2} \) |
| 19 | \( 1 + 9.13iT - 361T^{2} \) |
| 23 | \( 1 - 3.72iT - 529T^{2} \) |
| 29 | \( 1 + 1.12T + 841T^{2} \) |
| 31 | \( 1 + 57.0iT - 961T^{2} \) |
| 37 | \( 1 + 41.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 11.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 64.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 77.6T + 2.20e3T^{2} \) |
| 53 | \( 1 - 77.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 87.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 5.36iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 47.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 58.3T + 5.04e3T^{2} \) |
| 73 | \( 1 + 53.4T + 5.32e3T^{2} \) |
| 79 | \( 1 + 74.9T + 6.24e3T^{2} \) |
| 83 | \( 1 + 28.7T + 6.88e3T^{2} \) |
| 89 | \( 1 - 101. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 107.T + 9.40e3T^{2} \) |
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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
−9.965384965400306105272743159415, −9.427062048789261827823985236332, −8.242144446410561608460601941796, −7.43172804912680602022195099873, −6.20584942900529770082129057574, −5.09146637019041190839479314679, −4.39044353378403400296717838034, −2.85214884723837144042927067119, −1.92065099552823261105609080375, −0.35126887919271343168205936981,
1.96577030051275396376912226939, 2.99035238385690457226948408770, 4.88717301885787425579206541592, 5.34789384088139793861495850060, 6.30780423969280929714494405225, 7.19706258799485556368660979348, 8.365334185551361867312027919825, 8.857574608828734641048713454871, 10.00559582403438954462878565807, 10.51669179623673012553632435031