L(s) = 1 | + (4.60 + 2.65i)2-s + (0.153 + 5.19i)3-s + (10.1 + 17.5i)4-s + (−13.0 + 24.3i)6-s + (11.6 + 6.71i)7-s + 65.1i·8-s + (−26.9 + 1.59i)9-s + (23.4 − 40.6i)11-s + (−89.5 + 55.2i)12-s + (31.2 − 18.0i)13-s + (35.7 + 61.8i)14-s + (−92.1 + 159. i)16-s − 54.6i·17-s + (−128. − 64.3i)18-s − 111.·19-s + ⋯ |
L(s) = 1 | + (1.62 + 0.939i)2-s + (0.0295 + 0.999i)3-s + (1.26 + 2.19i)4-s + (−0.891 + 1.65i)6-s + (0.628 + 0.362i)7-s + 2.87i·8-s + (−0.998 + 0.0589i)9-s + (0.643 − 1.11i)11-s + (−2.15 + 1.33i)12-s + (0.667 − 0.385i)13-s + (0.681 + 1.18i)14-s + (−1.43 + 2.49i)16-s − 0.779i·17-s + (−1.68 − 0.841i)18-s − 1.34·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.836 - 0.547i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.836 - 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.33978 + 4.49668i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.33978 + 4.49668i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.153 - 5.19i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (-4.60 - 2.65i)T + (4 + 6.92i)T^{2} \) |
| 7 | \( 1 + (-11.6 - 6.71i)T + (171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (-23.4 + 40.6i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-31.2 + 18.0i)T + (1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 + 54.6iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 111.T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-31.1 + 17.9i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-29.0 + 50.3i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-147. - 256. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 53.0iT - 5.06e4T^{2} \) |
| 41 | \( 1 + (64.1 + 111. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (142. + 82.0i)T + (3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (76.0 + 43.9i)T + (5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 - 479. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (-317. - 550. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (24.0 - 41.5i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (25.0 - 14.4i)T + (1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 576.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 835. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + (101. - 176. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-402. - 232. i)T + (2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + 993.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (763. + 440. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−12.25295532819751289054455975331, −11.46519753415065972151699971757, −10.66903925095188480945276151239, −8.778890621751168732086529269061, −8.252096733075097880005069968742, −6.65616922590436029217586547575, −5.75977127372274041766048740454, −4.88141120198828302897984366287, −3.89347433220955960054633929511, −2.85824278398779058404534158212,
1.36391725151365134587599556523, 2.20861894458825557214975791032, 3.82254213423838761759639461303, 4.77077053909056228154250010788, 6.16978980580587250547125560452, 6.81297977677022330245668479967, 8.275866369120522468091087948825, 9.871275983569538329848912571804, 11.07804455666726033791755435141, 11.56782816149391953094197013422