| L(s) = 1 | + (−16 + 27.7i)2-s + (−511. − 886. i)4-s + (−1.88e3 − 3.27e3i)5-s + (2.65e4 − 4.59e4i)7-s + 3.27e4·8-s + 1.20e5·10-s + (−9.57e4 + 1.65e5i)11-s + (6.62e5 + 1.14e6i)13-s + (8.48e5 + 1.46e6i)14-s + (−5.24e5 + 9.08e5i)16-s − 7.39e6·17-s − 6.06e6·19-s + (−1.93e6 + 3.35e6i)20-s + (−3.06e6 − 5.30e6i)22-s + (−2.32e7 − 4.02e7i)23-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.270 − 0.468i)5-s + (0.596 − 1.03i)7-s + 0.353·8-s + 0.382·10-s + (−0.179 + 0.310i)11-s + (0.494 + 0.857i)13-s + (0.421 + 0.730i)14-s + (−0.125 + 0.216i)16-s − 1.26·17-s − 0.561·19-s + (−0.135 + 0.234i)20-s + (−0.126 − 0.219i)22-s + (−0.753 − 1.30i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.3758688692\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3758688692\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (16 - 27.7i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (1.88e3 + 3.27e3i)T + (-2.44e7 + 4.22e7i)T^{2} \) |
| 7 | \( 1 + (-2.65e4 + 4.59e4i)T + (-9.88e8 - 1.71e9i)T^{2} \) |
| 11 | \( 1 + (9.57e4 - 1.65e5i)T + (-1.42e11 - 2.47e11i)T^{2} \) |
| 13 | \( 1 + (-6.62e5 - 1.14e6i)T + (-8.96e11 + 1.55e12i)T^{2} \) |
| 17 | \( 1 + 7.39e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 6.06e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + (2.32e7 + 4.02e7i)T + (-4.76e14 + 8.25e14i)T^{2} \) |
| 29 | \( 1 + (-3.25e7 + 5.62e7i)T + (-6.10e15 - 1.05e16i)T^{2} \) |
| 31 | \( 1 + (-2.79e7 - 4.84e7i)T + (-1.27e16 + 2.20e16i)T^{2} \) |
| 37 | \( 1 + 2.00e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + (-2.86e8 - 4.95e8i)T + (-2.75e17 + 4.76e17i)T^{2} \) |
| 43 | \( 1 + (4.83e7 - 8.37e7i)T + (-4.64e17 - 8.04e17i)T^{2} \) |
| 47 | \( 1 + (-1.19e9 + 2.07e9i)T + (-1.23e18 - 2.14e18i)T^{2} \) |
| 53 | \( 1 + 3.14e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + (-1.10e9 - 1.90e9i)T + (-1.50e19 + 2.61e19i)T^{2} \) |
| 61 | \( 1 + (5.47e9 - 9.49e9i)T + (-2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (-5.31e9 - 9.20e9i)T + (-6.10e19 + 1.05e20i)T^{2} \) |
| 71 | \( 1 + 2.95e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 2.30e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + (-1.66e10 + 2.87e10i)T + (-3.73e20 - 6.47e20i)T^{2} \) |
| 83 | \( 1 + (1.17e10 - 2.02e10i)T + (-6.43e20 - 1.11e21i)T^{2} \) |
| 89 | \( 1 - 4.47e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + (1.23e10 - 2.13e10i)T + (-3.57e21 - 6.19e21i)T^{2} \) |
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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
−10.96775024487976793334416407443, −10.22662017972188142079963888897, −8.861234980914428187413740334486, −8.234901850840736090201183188476, −7.09738072610312396561316756489, −6.26533347105858448151001173358, −4.57586762177349341614068999499, −4.25069295863364529835785170797, −2.13325691825235278993820643763, −0.926188390860335078831428093979,
0.10104404199681342101545891320, 1.58860271062869529102810544261, 2.60131407007476396645722028471, 3.64945680595075859717667713903, 5.04243096854400252008257670128, 6.18659251428699222186755560263, 7.63095548025826678826435517426, 8.505712629025015401535843616507, 9.339676847288641031262670699612, 10.74261735746310665569803097811
