| L(s) = 1 | + 1.41i·2-s + (−0.991 + 2.83i)3-s − 2.00·4-s + (−6.36 + 3.67i)5-s + (−4.00 − 1.40i)6-s + (2.82 − 6.40i)7-s − 2.82i·8-s + (−7.03 − 5.61i)9-s + (−5.19 − 9.00i)10-s + (3.62 + 2.09i)11-s + (1.98 − 5.66i)12-s + (−0.596 + 1.03i)13-s + (9.05 + 3.99i)14-s + (−4.09 − 21.6i)15-s + 4.00·16-s + (−17.8 + 10.2i)17-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + (−0.330 + 0.943i)3-s − 0.500·4-s + (−1.27 + 0.735i)5-s + (−0.667 − 0.233i)6-s + (0.403 − 0.915i)7-s − 0.353i·8-s + (−0.781 − 0.623i)9-s + (−0.519 − 0.900i)10-s + (0.329 + 0.190i)11-s + (0.165 − 0.471i)12-s + (−0.0458 + 0.0794i)13-s + (0.647 + 0.285i)14-s + (−0.273 − 1.44i)15-s + 0.250·16-s + (−1.04 + 0.605i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.751 + 0.659i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.161649 - 0.429424i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.161649 - 0.429424i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 + (0.991 - 2.83i)T \) |
| 7 | \( 1 + (-2.82 + 6.40i)T \) |
| good | 5 | \( 1 + (6.36 - 3.67i)T + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (-3.62 - 2.09i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (0.596 - 1.03i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + (17.8 - 10.2i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (9.10 - 15.7i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (21.6 - 12.5i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (2.85 - 1.65i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + 51.9T + 961T^{2} \) |
| 37 | \( 1 + (-9.43 + 16.3i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-66.2 - 38.2i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-25.6 - 44.4i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 - 59.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-1.33 + 0.767i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + 32.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 111.T + 3.72e3T^{2} \) |
| 67 | \( 1 + 28.1T + 4.48e3T^{2} \) |
| 71 | \( 1 + 75.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-31.0 - 53.8i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + 67.1T + 6.24e3T^{2} \) |
| 83 | \( 1 + (64.7 - 37.3i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (95.3 + 55.0i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-14.0 - 24.3i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−14.43159052618940094123985739780, −12.72397210273656959513132834249, −11.32853473879961715018830550202, −10.84782038367860617268896207095, −9.612411376175592730984488153034, −8.235250302698070427017813541863, −7.31653042599476855496270502803, −6.09386184535603922127087471974, −4.34171496130651401214029659619, −3.80712771507195142606983146986,
0.32520906326826546632077365180, 2.27996738442029975546372173229, 4.24935210823206506237933463125, 5.52310764867213645424839126558, 7.17926603630295652811731241114, 8.409860514055597394621323743248, 8.971234013632683242003486477176, 11.01853080339670707323174449060, 11.62206398523102667302974934563, 12.35483105322166799588920692718
