| L(s) = 1 | + (1.5 + 2.59i)3-s + (−7.86 + 13.6i)5-s + (11.2 + 14.7i)7-s + (−4.5 + 7.79i)9-s + (−29.3 − 50.8i)11-s − 20.7·13-s − 47.2·15-s + (−21.4 − 37.1i)17-s + (−68.5 + 118. i)19-s + (−21.3 + 51.2i)21-s + (−14.5 + 25.1i)23-s + (−61.3 − 106. i)25-s − 27·27-s + 8.68·29-s + (101. + 175. i)31-s + ⋯ |
| L(s) = 1 | + (0.288 + 0.499i)3-s + (−0.703 + 1.21i)5-s + (0.606 + 0.794i)7-s + (−0.166 + 0.288i)9-s + (−0.804 − 1.39i)11-s − 0.442·13-s − 0.812·15-s + (−0.306 − 0.530i)17-s + (−0.828 + 1.43i)19-s + (−0.222 + 0.532i)21-s + (−0.131 + 0.228i)23-s + (−0.490 − 0.849i)25-s − 0.192·27-s + 0.0555·29-s + (0.586 + 1.01i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 - 0.325i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.945 - 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.171203 + 1.02207i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.171203 + 1.02207i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.5 - 2.59i)T \) |
| 7 | \( 1 + (-11.2 - 14.7i)T \) |
| good | 5 | \( 1 + (7.86 - 13.6i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (29.3 + 50.8i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 20.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + (21.4 + 37.1i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (68.5 - 118. i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (14.5 - 25.1i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 8.68T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-101. - 175. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-7.63 + 13.2i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 117.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 101.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (294. - 509. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (202. + 350. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (5.43 + 9.41i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-447. + 774. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-351. - 609. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 1.16e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-558. - 966. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (69.1 - 119. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 894.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-340. + 590. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 246.T + 9.12e5T^{2} \) |
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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
−12.67933769640600510657121783260, −11.43310204755041834368063992937, −10.95628871393973526743481041444, −9.912002936768215106527489370809, −8.445221186535511692877911606871, −7.88472893734689548554969980662, −6.39830764047956305269201500518, −5.14526098066255841488362114967, −3.57503008850049147508550607213, −2.55058055150555332589067167337,
0.44517792103480624313259135574, 2.10429690669693023939054804600, 4.27873419950512246466758963633, 4.92419917672185083916276025922, 6.89022785359064505055823001423, 7.81596816671132139606682129972, 8.537151382904928042110845409558, 9.784226262517940421090227330556, 11.00505734576592107480948787673, 12.12059137936050727097426284718
