| L(s) = 1 | − 1.41·2-s + 2.00·4-s + (−4.87 − 1.09i)5-s + 2.10i·7-s − 2.82·8-s + (6.89 + 1.55i)10-s − 13.3i·11-s + 23.1i·13-s − 2.97i·14-s + 4.00·16-s + 27.5·17-s − 15.5·19-s + (−9.75 − 2.19i)20-s + 18.8i·22-s − 14.9·23-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.500·4-s + (−0.975 − 0.219i)5-s + 0.300i·7-s − 0.353·8-s + (0.689 + 0.155i)10-s − 1.21i·11-s + 1.78i·13-s − 0.212i·14-s + 0.250·16-s + 1.61·17-s − 0.818·19-s + (−0.487 − 0.109i)20-s + 0.859i·22-s − 0.650·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.219 + 0.975i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.219 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5926282071\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5926282071\) |
| \(L(2)\) |
|
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 | \( 1 \) |
| 5 | \( 1 + (4.87 + 1.09i)T \) |
| good | 7 | \( 1 - 2.10iT - 49T^{2} \) |
| 11 | \( 1 + 13.3iT - 121T^{2} \) |
| 13 | \( 1 - 23.1iT - 169T^{2} \) |
| 17 | \( 1 - 27.5T + 289T^{2} \) |
| 19 | \( 1 + 15.5T + 361T^{2} \) |
| 23 | \( 1 + 14.9T + 529T^{2} \) |
| 29 | \( 1 - 3.28iT - 841T^{2} \) |
| 31 | \( 1 + 9.84T + 961T^{2} \) |
| 37 | \( 1 + 33.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 10.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 43.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 36.0T + 2.20e3T^{2} \) |
| 53 | \( 1 + 53.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + 101. iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 0.101T + 3.72e3T^{2} \) |
| 67 | \( 1 + 55.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 79.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 20.8iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 1.48T + 6.24e3T^{2} \) |
| 83 | \( 1 + 9.93T + 6.88e3T^{2} \) |
| 89 | \( 1 + 152. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 38.7iT - 9.40e3T^{2} \) |
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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
−9.646199148666816228854024666489, −8.831292346362029187982954063356, −8.265122235752223715622493551819, −7.40876710952761366981220169016, −6.47951845862210116764953140440, −5.50010774113021978161510572612, −4.15718640487829594867530970113, −3.28353810414361869494644280635, −1.76527904153797723388444665167, −0.29922939271560699308318680658,
1.09289498807599286235476830328, 2.74271199135938614162134121726, 3.73002163334579307381391539187, 4.90795153122672122669026905441, 6.07591376586963383033590044619, 7.20388813547051766361794557069, 7.84269308605872684617594592261, 8.282251071632274233377151384297, 9.643586805320084840632294660875, 10.27803926718571652548984492624
