| 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 + (−3.71 − 14.5i)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 + (−0.247 − 0.968i)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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.490 + 0.871i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.490 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1864276322\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1864276322\) |
| \(L(2)\) |
|
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
−10.00565925771190169188784999498, −8.745005524771598960024874007112, −8.147680288770309720068299717425, −7.32720305857654828146257822293, −5.88371635493392609670459311823, −5.45880453093034310783519271923, −4.19678307349072129640951841981, −3.46914610196060889685433224929, −2.40016813776343861740542877379, −0.07609169061641246220724329056,
0.933158261656587255768078277434, 2.42671730786585012133926367695, 3.61798877284069687178607915353, 4.79925067205876648269514168055, 5.63150331670440651100100579055, 6.76748396276214691261271446454, 7.63210121510576929387985896270, 7.986643945327849624585913620900, 8.866727626225721376575171195887, 10.15178314437129577206612425811
