| L(s) = 1 | + (1 − 1.73i)2-s + (−1.99 − 3.46i)4-s + (−10.5 − 18.1i)5-s + (−4 + 6.92i)7-s − 7.99·8-s − 42·10-s + (−18 + 31.1i)11-s + (24.5 + 42.4i)13-s + (7.99 + 13.8i)14-s + (−8 + 13.8i)16-s + 21·17-s − 112·19-s + (−42 + 72.7i)20-s + (36 + 62.3i)22-s + (−90 − 155. i)23-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.939 − 1.62i)5-s + (−0.215 + 0.374i)7-s − 0.353·8-s − 1.32·10-s + (−0.493 + 0.854i)11-s + (0.522 + 0.905i)13-s + (0.152 + 0.264i)14-s + (−0.125 + 0.216i)16-s + 0.299·17-s − 1.35·19-s + (−0.469 + 0.813i)20-s + (0.348 + 0.604i)22-s + (−0.815 − 1.41i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.175588 + 0.482425i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.175588 + 0.482425i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1 + 1.73i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (10.5 + 18.1i)T + (-62.5 + 108. i)T^{2} \) |
| 7 | \( 1 + (4 - 6.92i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (18 - 31.1i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-24.5 - 42.4i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 - 21T + 4.91e3T^{2} \) |
| 19 | \( 1 + 112T + 6.85e3T^{2} \) |
| 23 | \( 1 + (90 + 155. i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-67.5 + 116. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (154 + 266. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-21 - 36.3i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (10 - 17.3i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (42 - 72.7i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + 174T + 1.48e5T^{2} \) |
| 59 | \( 1 + (252 + 436. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-192.5 + 333. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (136 + 235. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 888T + 3.57e5T^{2} \) |
| 73 | \( 1 - 371T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-326 + 564. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (42 - 72.7i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 - 21T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-623 + 1.07e3i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−12.09879322731518101140020042955, −11.04557357365808662308218985747, −9.681820054894133040611097044786, −8.771157109736930641407290157728, −7.88652863805322857955551795292, −6.12150945478284638631593709068, −4.67159433361679494572306995020, −4.09332241226495904768042348001, −1.99816042053293760374227000223, −0.20703177669386046574587578993,
3.08025880762221961439731613642, 3.81956958809398824250831255985, 5.68476367735195585342858244949, 6.74701687838380317746794082205, 7.62743829993428817961307669180, 8.497735024432464211696807733753, 10.38651324377266630996603287847, 10.86580878045095246601659759591, 12.00852182784306743757573320395, 13.22243735946447766465928615193
