| L(s) = 1 | + (−122. + 70.7i)2-s + (−1.80e3 − 1.22e3i)3-s + (1.83e3 − 3.17e3i)4-s + (−1.01e5 − 5.87e4i)5-s + (3.08e5 + 2.27e4i)6-s + (−5.25e5 − 9.09e5i)7-s − 1.80e6i·8-s + (1.75e6 + 4.44e6i)9-s + 1.66e7·10-s + (−2.18e7 + 1.26e7i)11-s + (−7.21e6 + 3.48e6i)12-s + (−3.00e7 + 5.20e7i)13-s + (1.28e8 + 7.43e7i)14-s + (1.11e8 + 2.31e8i)15-s + (1.57e8 + 2.72e8i)16-s − 3.83e8i·17-s + ⋯ |
| L(s) = 1 | + (−0.957 + 0.553i)2-s + (−0.826 − 0.562i)3-s + (0.111 − 0.193i)4-s + (−1.30 − 0.751i)5-s + (1.10 + 0.0812i)6-s + (−0.637 − 1.10i)7-s − 0.858i·8-s + (0.367 + 0.929i)9-s + 1.66·10-s + (−1.12 + 0.648i)11-s + (−0.201 + 0.0973i)12-s + (−0.479 + 0.830i)13-s + (1.22 + 0.705i)14-s + (0.654 + 1.35i)15-s + (0.586 + 1.01i)16-s − 0.933i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.475 - 0.879i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (0.475 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{15}{2})\) |
\(\approx\) |
\(0.108124 + 0.0644270i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.108124 + 0.0644270i\) |
| \(L(8)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.80e3 + 1.22e3i)T \) |
| good | 2 | \( 1 + (122. - 70.7i)T + (8.19e3 - 1.41e4i)T^{2} \) |
| 5 | \( 1 + (1.01e5 + 5.87e4i)T + (3.05e9 + 5.28e9i)T^{2} \) |
| 7 | \( 1 + (5.25e5 + 9.09e5i)T + (-3.39e11 + 5.87e11i)T^{2} \) |
| 11 | \( 1 + (2.18e7 - 1.26e7i)T + (1.89e14 - 3.28e14i)T^{2} \) |
| 13 | \( 1 + (3.00e7 - 5.20e7i)T + (-1.96e15 - 3.40e15i)T^{2} \) |
| 17 | \( 1 + 3.83e8iT - 1.68e17T^{2} \) |
| 19 | \( 1 - 8.06e8T + 7.99e17T^{2} \) |
| 23 | \( 1 + (4.24e9 + 2.45e9i)T + (5.79e18 + 1.00e19i)T^{2} \) |
| 29 | \( 1 + (1.35e9 - 7.83e8i)T + (1.48e20 - 2.57e20i)T^{2} \) |
| 31 | \( 1 + (-7.81e9 + 1.35e10i)T + (-3.78e20 - 6.55e20i)T^{2} \) |
| 37 | \( 1 - 4.75e10T + 9.01e21T^{2} \) |
| 41 | \( 1 + (-2.70e10 - 1.56e10i)T + (1.89e22 + 3.28e22i)T^{2} \) |
| 43 | \( 1 + (4.58e10 + 7.94e10i)T + (-3.69e22 + 6.39e22i)T^{2} \) |
| 47 | \( 1 + (4.80e11 - 2.77e11i)T + (1.28e23 - 2.22e23i)T^{2} \) |
| 53 | \( 1 + 9.95e11iT - 1.37e24T^{2} \) |
| 59 | \( 1 + (-1.03e12 - 5.97e11i)T + (3.09e24 + 5.36e24i)T^{2} \) |
| 61 | \( 1 + (-1.22e12 - 2.12e12i)T + (-4.93e24 + 8.55e24i)T^{2} \) |
| 67 | \( 1 + (3.05e12 - 5.28e12i)T + (-1.83e25 - 3.18e25i)T^{2} \) |
| 71 | \( 1 - 1.63e12iT - 8.27e25T^{2} \) |
| 73 | \( 1 + 3.24e12T + 1.22e26T^{2} \) |
| 79 | \( 1 + (1.75e13 + 3.03e13i)T + (-1.84e26 + 3.19e26i)T^{2} \) |
| 83 | \( 1 + (-2.23e13 + 1.29e13i)T + (3.68e26 - 6.37e26i)T^{2} \) |
| 89 | \( 1 - 3.18e13iT - 1.95e27T^{2} \) |
| 97 | \( 1 + (2.38e13 + 4.13e13i)T + (-3.26e27 + 5.65e27i)T^{2} \) |
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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
−17.85494743118612911732025618943, −16.36831372239973774134262684964, −16.12849834133567370639924751365, −13.16043512806952339646397620732, −11.91345914670153037303960006010, −9.978225780010060558759172567385, −7.85847957821126017903113644463, −7.09432588900736308827967497705, −4.42640340294525115132897653956, −0.59789637311495805519102347126,
0.17233193538059957599602472101, 3.10636184855560173907147564590, 5.61709151499450586707142172188, 8.057744496551075092708433626784, 9.920608918702368786451307647326, 11.05764225060586268989554224435, 12.17284744589737912964946178995, 15.15582530913080262004113571501, 16.01973319092335850786809801792, 17.94073593163326538043094513427
