| L(s) = 1 | + (0.707 + 1.22i)2-s + (−1.5 − 0.866i)3-s + (−0.999 + 1.73i)4-s + (1.93 − 1.11i)5-s − 2.44i·6-s + (−6.38 + 2.86i)7-s − 2.82·8-s + (1.5 + 2.59i)9-s + (2.73 + 1.58i)10-s + (−9.98 + 17.2i)11-s + (2.99 − 1.73i)12-s + 3.49i·13-s + (−8.02 − 5.80i)14-s − 3.87·15-s + (−2.00 − 3.46i)16-s + (15.7 + 9.12i)17-s + ⋯ |
| L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.5 − 0.288i)3-s + (−0.249 + 0.433i)4-s + (0.387 − 0.223i)5-s − 0.408i·6-s + (−0.912 + 0.408i)7-s − 0.353·8-s + (0.166 + 0.288i)9-s + (0.273 + 0.158i)10-s + (−0.907 + 1.57i)11-s + (0.249 − 0.144i)12-s + 0.269i·13-s + (−0.572 − 0.414i)14-s − 0.258·15-s + (−0.125 − 0.216i)16-s + (0.929 + 0.536i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.877 - 0.478i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.877 - 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.219331 + 0.860106i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.219331 + 0.860106i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.707 - 1.22i)T \) |
| 3 | \( 1 + (1.5 + 0.866i)T \) |
| 5 | \( 1 + (-1.93 + 1.11i)T \) |
| 7 | \( 1 + (6.38 - 2.86i)T \) |
| good | 11 | \( 1 + (9.98 - 17.2i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 - 3.49iT - 169T^{2} \) |
| 17 | \( 1 + (-15.7 - 9.12i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (21.3 - 12.3i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (12.5 + 21.8i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + 53.1T + 841T^{2} \) |
| 31 | \( 1 + (-26.0 - 15.0i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-23.3 - 40.5i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + 31.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 64.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + (24.3 - 14.0i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-32.4 + 56.1i)T + (-1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-86.7 - 50.0i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-6.94 + 4.01i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (8.13 - 14.0i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 107.T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-44.7 - 25.8i)T + (2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-10.9 - 19.0i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 - 0.417iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (96.3 - 55.6i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 - 74.2iT - 9.40e3T^{2} \) |
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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
−12.70390826521165800455278515055, −12.05257095902233071580156075087, −10.38750955171896037443370654001, −9.729936549011592948709808022104, −8.383991469713973051088022526565, −7.27985553522603690102482954993, −6.27649035787404754084088510213, −5.43337018722522655505026852016, −4.17528514010419676165117201370, −2.26795925201286165099887601527,
0.44930254686225314879534086806, 2.78921908763615392791861669542, 3.87778679824886224440904460885, 5.51484203064957522984751049815, 6.10496362434769186844179033355, 7.61523916762619274386852759778, 9.135152754782289818045024866444, 10.03784309055969210236532996366, 10.82432610902628625049418228443, 11.52879446814806444890940613666
