| 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 | \( 1 + 74.2iT - 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.52879446814806444890940613666, −10.82432610902628625049418228443, −10.03784309055969210236532996366, −9.135152754782289818045024866444, −7.61523916762619274386852759778, −6.10496362434769186844179033355, −5.51484203064957522984751049815, −3.87778679824886224440904460885, −2.78921908763615392791861669542, −0.44930254686225314879534086806,
2.26795925201286165099887601527, 4.17528514010419676165117201370, 5.43337018722522655505026852016, 6.27649035787404754084088510213, 7.27985553522603690102482954993, 8.383991469713973051088022526565, 9.729936549011592948709808022104, 10.38750955171896037443370654001, 12.05257095902233071580156075087, 12.70390826521165800455278515055
