| L(s) = 1 | + (−1.38 − 7.87i)2-s + (−44.6 − 13.9i)3-s + (−60.1 + 21.8i)4-s + (200. + 168. i)5-s + (−47.6 + 371. i)6-s + (−437. − 159. i)7-s + (256 + 443. i)8-s + (1.79e3 + 1.24e3i)9-s + (1.04e3 − 1.81e3i)10-s + (4.95e3 − 4.15e3i)11-s + (2.98e3 − 140. i)12-s + (−2.29e3 + 1.30e4i)13-s + (−647. + 3.67e3i)14-s + (−6.61e3 − 1.03e4i)15-s + (3.13e3 − 2.63e3i)16-s + (1.28e4 − 2.22e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 − 0.696i)2-s + (−0.954 − 0.297i)3-s + (−0.469 + 0.171i)4-s + (0.717 + 0.602i)5-s + (−0.0900 + 0.701i)6-s + (−0.482 − 0.175i)7-s + (0.176 + 0.306i)8-s + (0.822 + 0.568i)9-s + (0.331 − 0.573i)10-s + (1.12 − 0.942i)11-s + (0.499 − 0.0234i)12-s + (−0.289 + 1.64i)13-s + (−0.0630 + 0.357i)14-s + (−0.506 − 0.788i)15-s + (0.191 − 0.160i)16-s + (0.634 − 1.09i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.319 + 0.947i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.319 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.662721 - 0.922508i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.662721 - 0.922508i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.38 + 7.87i)T \) |
| 3 | \( 1 + (44.6 + 13.9i)T \) |
| good | 5 | \( 1 + (-200. - 168. i)T + (1.35e4 + 7.69e4i)T^{2} \) |
| 7 | \( 1 + (437. + 159. i)T + (6.30e5 + 5.29e5i)T^{2} \) |
| 11 | \( 1 + (-4.95e3 + 4.15e3i)T + (3.38e6 - 1.91e7i)T^{2} \) |
| 13 | \( 1 + (2.29e3 - 1.30e4i)T + (-5.89e7 - 2.14e7i)T^{2} \) |
| 17 | \( 1 + (-1.28e4 + 2.22e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (2.43e4 + 4.21e4i)T + (-4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (7.21e4 - 2.62e4i)T + (2.60e9 - 2.18e9i)T^{2} \) |
| 29 | \( 1 + (760. + 4.31e3i)T + (-1.62e10 + 5.89e9i)T^{2} \) |
| 31 | \( 1 + (-2.58e5 + 9.39e4i)T + (2.10e10 - 1.76e10i)T^{2} \) |
| 37 | \( 1 + (-7.98e4 + 1.38e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + (-2.76e4 + 1.56e5i)T + (-1.83e11 - 6.66e10i)T^{2} \) |
| 43 | \( 1 + (-7.18e5 + 6.02e5i)T + (4.72e10 - 2.67e11i)T^{2} \) |
| 47 | \( 1 + (-2.69e5 - 9.82e4i)T + (3.88e11 + 3.25e11i)T^{2} \) |
| 53 | \( 1 - 9.97e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + (3.77e5 + 3.16e5i)T + (4.32e11 + 2.45e12i)T^{2} \) |
| 61 | \( 1 + (-1.78e6 - 6.49e5i)T + (2.40e12 + 2.02e12i)T^{2} \) |
| 67 | \( 1 + (4.26e5 - 2.41e6i)T + (-5.69e12 - 2.07e12i)T^{2} \) |
| 71 | \( 1 + (-1.08e6 + 1.88e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + (2.08e6 + 3.60e6i)T + (-5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (6.39e5 + 3.62e6i)T + (-1.80e13 + 6.56e12i)T^{2} \) |
| 83 | \( 1 + (-1.67e5 - 9.51e5i)T + (-2.54e13 + 9.28e12i)T^{2} \) |
| 89 | \( 1 + (3.23e6 + 5.60e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + (7.44e5 - 6.24e5i)T + (1.40e13 - 7.95e13i)T^{2} \) |
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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
−13.57897742224417242084162464330, −11.98186868571027074102168747213, −11.39085570606140281369355015937, −10.13323486967555031847810374570, −9.155719701339995078407889948163, −6.98326193350902935938105116506, −6.06146233431700517597089312584, −4.23841170888534505640605589153, −2.26303392356194376559473253224, −0.60259903491006689431996296571,
1.22620080481701730453967909182, 4.17723251839368369439497187244, 5.65328431677386865752180355218, 6.35786924103461251673971427740, 8.117993940366492252642908263701, 9.739840353073455648257041360450, 10.24724028897343483223849361867, 12.33767454660898585039358807261, 12.75100667931687306521429214301, 14.49299202216539874270751438319
