| L(s) = 1 | + (−2.5 + 4.33i)2-s + (−8.50 − 14.7i)4-s + 45.0·8-s + (−34 − 58.8i)11-s + (−44.5 + 77.0i)16-s + 340·22-s + (−20 + 34.6i)23-s + (62.5 + 108. i)25-s + 166·29-s + (−42.5 − 73.6i)32-s + (−225 + 389. i)37-s − 180·43-s + (−578. + 1.00e3i)44-s + (−100. − 173. i)46-s − 625·50-s + ⋯ |
| L(s) = 1 | + (−0.883 + 1.53i)2-s + (−1.06 − 1.84i)4-s + 1.98·8-s + (−0.931 − 1.61i)11-s + (−0.695 + 1.20i)16-s + 3.29·22-s + (−0.181 + 0.314i)23-s + (0.5 + 0.866i)25-s + 1.06·29-s + (−0.234 − 0.406i)32-s + (−0.999 + 1.73i)37-s − 0.638·43-s + (−1.98 + 3.43i)44-s + (−0.320 − 0.555i)46-s − 1.76·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5855890011\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5855890011\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + (2.5 - 4.33i)T + (-4 - 6.92i)T^{2} \) |
| 5 | \( 1 + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (34 + 58.8i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 2.19e3T^{2} \) |
| 17 | \( 1 + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (20 - 34.6i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 166T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (225 - 389. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 6.89e4T^{2} \) |
| 43 | \( 1 + 180T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-295 - 510. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-370 - 640. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 688T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-692 + 1.19e3i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 5.71e5T^{2} \) |
| 89 | \( 1 + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 9.12e5T^{2} \) |
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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
−10.72488346238157930997773090150, −10.05913350975079701122932665626, −8.889780367959263699654933175021, −8.390599288122067433831491335047, −7.56267671468386055070745546481, −6.57075982543942919019653135865, −5.72842454974206573142089391840, −4.93235381803303617779544136860, −3.13981837437279177474721732602, −0.991250718959446163854458220873,
0.31981083005606228011148247347, 1.87588901411534692339525092058, 2.69183637303999692990359341430, 4.04039582634859740243095377758, 5.06251122322544947950230412996, 6.85999101899748916062975684564, 7.88750820645534726855723970240, 8.691118422527247735811299287186, 9.710361886567874244111428662946, 10.27649055342522064769704212857
