| L(s) = 1 | + (−28.4 − 14.6i)2-s − 233. i·3-s + (596. + 832. i)4-s + (−2.18e3 + 2.23e3i)5-s + (−3.41e3 + 6.64e3i)6-s + (4.25e3 + 4.25e3i)7-s + (−4.79e3 − 3.24e4i)8-s + 4.56e3·9-s + (9.49e4 − 3.14e4i)10-s + (−2.11e5 + 2.11e5i)11-s + (1.94e5 − 1.39e5i)12-s + 3.98e5i·13-s + (−5.88e4 − 1.83e5i)14-s + (5.20e5 + 5.10e5i)15-s + (−3.37e5 + 9.92e5i)16-s + (−1.05e6 + 1.05e6i)17-s + ⋯ |
| L(s) = 1 | + (−0.889 − 0.457i)2-s − 0.960i·3-s + (0.582 + 0.813i)4-s + (−0.700 + 0.713i)5-s + (−0.439 + 0.854i)6-s + (0.253 + 0.253i)7-s + (−0.146 − 0.989i)8-s + 0.0772·9-s + (0.949 − 0.314i)10-s + (−1.31 + 1.31i)11-s + (0.781 − 0.559i)12-s + 1.07i·13-s + (−0.109 − 0.340i)14-s + (0.685 + 0.672i)15-s + (−0.322 + 0.946i)16-s + (−0.739 + 0.739i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 + 0.301i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.953 + 0.301i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.0286516 - 0.185636i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0286516 - 0.185636i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (28.4 + 14.6i)T \) |
| 5 | \( 1 + (2.18e3 - 2.23e3i)T \) |
| good | 3 | \( 1 + 233. iT - 5.90e4T^{2} \) |
| 7 | \( 1 + (-4.25e3 - 4.25e3i)T + 2.82e8iT^{2} \) |
| 11 | \( 1 + (2.11e5 - 2.11e5i)T - 2.59e10iT^{2} \) |
| 13 | \( 1 - 3.98e5iT - 1.37e11T^{2} \) |
| 17 | \( 1 + (1.05e6 - 1.05e6i)T - 2.01e12iT^{2} \) |
| 19 | \( 1 + (-1.79e6 + 1.79e6i)T - 6.13e12iT^{2} \) |
| 23 | \( 1 + (2.99e6 - 2.99e6i)T - 4.14e13iT^{2} \) |
| 29 | \( 1 + (-1.55e7 + 1.55e7i)T - 4.20e14iT^{2} \) |
| 31 | \( 1 + 1.02e7T + 8.19e14T^{2} \) |
| 37 | \( 1 - 9.48e7iT - 4.80e15T^{2} \) |
| 41 | \( 1 + 1.88e8iT - 1.34e16T^{2} \) |
| 43 | \( 1 - 1.96e8T + 2.16e16T^{2} \) |
| 47 | \( 1 + (5.83e7 - 5.83e7i)T - 5.25e16iT^{2} \) |
| 53 | \( 1 + 7.63e8T + 1.74e17T^{2} \) |
| 59 | \( 1 + (-2.64e8 - 2.64e8i)T + 5.11e17iT^{2} \) |
| 61 | \( 1 + (3.97e8 + 3.97e8i)T + 7.13e17iT^{2} \) |
| 67 | \( 1 + 9.93e8T + 1.82e18T^{2} \) |
| 71 | \( 1 - 1.63e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 + (-1.99e9 + 1.99e9i)T - 4.29e18iT^{2} \) |
| 79 | \( 1 + 4.43e8iT - 9.46e18T^{2} \) |
| 83 | \( 1 - 2.39e9iT - 1.55e19T^{2} \) |
| 89 | \( 1 + 7.16e9T + 3.11e19T^{2} \) |
| 97 | \( 1 + (6.52e9 - 6.52e9i)T - 7.37e19iT^{2} \) |
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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
−11.82666774611405080602531135313, −10.78549719406056660972392699260, −9.686263990882875955217934398735, −8.172142597385521695758921108402, −7.38741825971665420246578243726, −6.65051647179870281807395726763, −4.32189306979175988352450842109, −2.56668274158830583903361825790, −1.74630449729770407033779587220, −0.082270730130493900311455574828,
0.938851222553098929686532407506, 3.08206064813833310400133694491, 4.73408708498441095259816157305, 5.66083347432653735231598263746, 7.58441592754679275244939728428, 8.298772512975175964685232579541, 9.405463610486907038352301587755, 10.57529564314920699761536437337, 11.15097064085322068671696866589, 12.74324006159798087472018570554
