L(s) = 1 | + (1 − 1.73i)2-s + (0.5 + 0.866i)3-s + (−1.99 − 3.46i)4-s + (−3.5 + 6.06i)5-s + 1.99·6-s + (−10 + 15.5i)7-s − 7.99·8-s + (13 − 22.5i)9-s + (7 + 12.1i)10-s + (−17.5 − 30.3i)11-s + (1.99 − 3.46i)12-s + 66·13-s + (17 + 32.9i)14-s − 7·15-s + (−8 + 13.8i)16-s + (−29.5 − 51.0i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (0.0962 + 0.166i)3-s + (−0.249 − 0.433i)4-s + (−0.313 + 0.542i)5-s + 0.136·6-s + (−0.539 + 0.841i)7-s − 0.353·8-s + (0.481 − 0.833i)9-s + (0.221 + 0.383i)10-s + (−0.479 − 0.830i)11-s + (0.0481 − 0.0833i)12-s + 1.40·13-s + (0.324 + 0.628i)14-s − 0.120·15-s + (−0.125 + 0.216i)16-s + (−0.420 − 0.728i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 + 0.485i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.874 + 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.03971 - 0.269364i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.03971 - 0.269364i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 + 1.73i)T \) |
| 7 | \( 1 + (10 - 15.5i)T \) |
good | 3 | \( 1 + (-0.5 - 0.866i)T + (-13.5 + 23.3i)T^{2} \) |
| 5 | \( 1 + (3.5 - 6.06i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (17.5 + 30.3i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 66T + 2.19e3T^{2} \) |
| 17 | \( 1 + (29.5 + 51.0i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (68.5 - 118. i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-3.5 + 6.06i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 106T + 2.43e4T^{2} \) |
| 31 | \( 1 + (37.5 + 64.9i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (5.5 - 9.52i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 498T + 6.89e4T^{2} \) |
| 43 | \( 1 - 260T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-85.5 + 148. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-208.5 - 361. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-8.5 - 14.7i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (25.5 - 44.1i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (219.5 + 380. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 784T + 3.57e5T^{2} \) |
| 73 | \( 1 + (147.5 + 255. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-247.5 + 428. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 932T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-436.5 + 756. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 290T + 9.12e5T^{2} \) |
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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
−18.84861829834191611688641591318, −18.42395395886323971363820162841, −16.03861877898491807075026006561, −15.00722107260579078507327498172, −13.38296499294512487966267915579, −11.97993937386865071289635495549, −10.54641437726097788540688220612, −8.839065203126092887000071158281, −6.17419372141557392180812905306, −3.44148627176067932895503895911,
4.45809912835375765151201974082, 6.86227351184222756791851788691, 8.452930461734506816688691865851, 10.60580380214991759514489550893, 12.82745157815110757261047381609, 13.54746730171127226533518152529, 15.47552069754499605822576797552, 16.38042466698445010529767813229, 17.73376855449346056640164989882, 19.36659132660275986647232450062