| L(s) = 1 | + (0.619 − 2.93i)3-s + (−4.48 − 2.21i)5-s − 4.63i·7-s + (−8.23 − 3.63i)9-s − 15.0·11-s − 3.35·13-s + (−9.27 + 11.7i)15-s − 11.1·17-s + 22.7i·19-s + (−13.5 − 2.87i)21-s + 36.1·23-s + (15.2 + 19.8i)25-s + (−15.7 + 21.9i)27-s − 21.5·29-s + 28.3·31-s + ⋯ |
| L(s) = 1 | + (0.206 − 0.978i)3-s + (−0.896 − 0.442i)5-s − 0.661i·7-s + (−0.914 − 0.404i)9-s − 1.37·11-s − 0.258·13-s + (−0.618 + 0.785i)15-s − 0.655·17-s + 1.19i·19-s + (−0.647 − 0.136i)21-s + 1.57·23-s + (0.608 + 0.793i)25-s + (−0.584 + 0.811i)27-s − 0.743·29-s + 0.915·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.393 - 0.919i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.393 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2705832629\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2705832629\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.619 + 2.93i)T \) |
| 5 | \( 1 + (4.48 + 2.21i)T \) |
| good | 7 | \( 1 + 4.63iT - 49T^{2} \) |
| 11 | \( 1 + 15.0T + 121T^{2} \) |
| 13 | \( 1 + 3.35T + 169T^{2} \) |
| 17 | \( 1 + 11.1T + 289T^{2} \) |
| 19 | \( 1 - 22.7iT - 361T^{2} \) |
| 23 | \( 1 - 36.1T + 529T^{2} \) |
| 29 | \( 1 + 21.5T + 841T^{2} \) |
| 31 | \( 1 - 28.3T + 961T^{2} \) |
| 37 | \( 1 - 69.5T + 1.36e3T^{2} \) |
| 41 | \( 1 + 62.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 55.2T + 1.84e3T^{2} \) |
| 47 | \( 1 + 36.5T + 2.20e3T^{2} \) |
| 53 | \( 1 + 29.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 8.07T + 3.48e3T^{2} \) |
| 61 | \( 1 - 10.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 91.8T + 4.48e3T^{2} \) |
| 71 | \( 1 - 79.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 59.0iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 86.3T + 6.24e3T^{2} \) |
| 83 | \( 1 - 10.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 17.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 139. iT - 9.40e3T^{2} \) |
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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
−10.03064745500041757323694929027, −8.856030671459501733555087067142, −8.138228483486242539574484982588, −7.52779655897566957386509914631, −6.90126617244336704907214178831, −5.67380748865761664849828120371, −4.71917034568317338722195120145, −3.57579211374292411300457252015, −2.50543801368464227440471790755, −1.05091516239635269620542599307,
0.095999208668190341000093210292, 2.67296549375804512860606382712, 3.03435216150969533208813283465, 4.54373529594069824242124401849, 4.95002871814379861190407972287, 6.16626845525015307305982735881, 7.30836548281738146546191149207, 8.122614740365550638710531664880, 8.854989540872721493819782319205, 9.661365625515766848760330123024
