| L(s) = 1 | + (4.89 + 2.82i)2-s + (15.9 + 27.7i)4-s + (−29.3 + 16.9i)5-s + (102.5 − 177. i)7-s + 181. i·8-s − 192·10-s + (−911. − 526. i)11-s + (1.02e3 + 1.76e3i)13-s + (1.00e3 − 579. i)14-s + (−512. + 886. i)16-s + 8.24e3i·17-s − 1.50e3·19-s + (−940. − 543. i)20-s + (−2.97e3 − 5.15e3i)22-s + (6.14e3 − 3.54e3i)23-s + ⋯ |
| L(s) = 1 | + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.235 + 0.135i)5-s + (0.298 − 0.517i)7-s + 0.353i·8-s − 0.192·10-s + (−0.684 − 0.395i)11-s + (0.464 + 0.804i)13-s + (0.365 − 0.211i)14-s + (−0.125 + 0.216i)16-s + 1.67i·17-s − 0.218·19-s + (−0.117 − 0.0678i)20-s + (−0.279 − 0.484i)22-s + (0.504 − 0.291i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.776100357\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.776100357\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-4.89 - 2.82i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (29.3 - 16.9i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 7 | \( 1 + (-102.5 + 177. i)T + (-5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (911. + 526. i)T + (8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 + (-1.02e3 - 1.76e3i)T + (-2.41e6 + 4.18e6i)T^{2} \) |
| 17 | \( 1 - 8.24e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 1.50e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + (-6.14e3 + 3.54e3i)T + (7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 + (2.46e4 + 1.42e4i)T + (2.97e8 + 5.15e8i)T^{2} \) |
| 31 | \( 1 + (-1.74e4 - 3.03e4i)T + (-4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + 5.76e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + (1.16e5 - 6.71e4i)T + (2.37e9 - 4.11e9i)T^{2} \) |
| 43 | \( 1 + (-3.12e4 + 5.41e4i)T + (-3.16e9 - 5.47e9i)T^{2} \) |
| 47 | \( 1 + (-4.49e4 - 2.59e4i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 - 7.71e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 + (3.21e5 - 1.85e5i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (-3.06e4 + 5.30e4i)T + (-2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (3.38e4 + 5.86e4i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 - 5.09e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 4.23e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + (-3.53e5 + 6.12e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + (7.45e5 + 4.30e5i)T + (1.63e11 + 2.83e11i)T^{2} \) |
| 89 | \( 1 - 6.67e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (2.63e5 - 4.55e5i)T + (-4.16e11 - 7.21e11i)T^{2} \) |
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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
−12.26291342258923211335071358450, −11.13893716606079345117653653154, −10.44302048224211301127487852663, −8.804184655219194342051979750086, −7.88296931412457512915535770948, −6.79479001759124514851492605792, −5.70300270716701104192007447308, −4.38894370929594914401737786823, −3.38267251463268443258639927200, −1.64867296824593857901583009458,
0.39874791575154027116361606053, 2.12096955723894106122197438597, 3.32182845732779314006088055195, 4.82027391080730171364426587485, 5.60161403806600794908995456359, 7.08728345148794629905124755075, 8.213591273346551986322033029412, 9.452514360989537999470636965583, 10.55013533487882805502995373266, 11.52945011098373338980528971718
