| L(s) = 1 | + 0.638·3-s − 9.06i·7-s − 8.59·9-s + (10.0 − 4.36i)11-s + 10.4i·13-s + 3.39i·17-s − 28.0i·19-s − 5.79i·21-s − 18.8·23-s − 11.2·27-s + 22.2i·29-s + 0.00744·31-s + (6.44 − 2.79i)33-s − 8.96·37-s + 6.66i·39-s + ⋯ |
| L(s) = 1 | + 0.212·3-s − 1.29i·7-s − 0.954·9-s + (0.917 − 0.397i)11-s + 0.802i·13-s + 0.199i·17-s − 1.47i·19-s − 0.275i·21-s − 0.818·23-s − 0.416·27-s + 0.766i·29-s + 0.000240·31-s + (0.195 − 0.0845i)33-s − 0.242·37-s + 0.170i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.917 + 0.397i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.917 + 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8502174465\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8502174465\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + (-10.0 + 4.36i)T \) |
| good | 3 | \( 1 - 0.638T + 9T^{2} \) |
| 7 | \( 1 + 9.06iT - 49T^{2} \) |
| 13 | \( 1 - 10.4iT - 169T^{2} \) |
| 17 | \( 1 - 3.39iT - 289T^{2} \) |
| 19 | \( 1 + 28.0iT - 361T^{2} \) |
| 23 | \( 1 + 18.8T + 529T^{2} \) |
| 29 | \( 1 - 22.2iT - 841T^{2} \) |
| 31 | \( 1 - 0.00744T + 961T^{2} \) |
| 37 | \( 1 + 8.96T + 1.36e3T^{2} \) |
| 41 | \( 1 - 16.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 81.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 59.1T + 2.20e3T^{2} \) |
| 53 | \( 1 + 60.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + 81.5T + 3.48e3T^{2} \) |
| 61 | \( 1 - 114. iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 99.1T + 4.48e3T^{2} \) |
| 71 | \( 1 - 70.3T + 5.04e3T^{2} \) |
| 73 | \( 1 + 16.4iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 118. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 30.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 4.02T + 7.92e3T^{2} \) |
| 97 | \( 1 + 101.T + 9.40e3T^{2} \) |
| 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
−9.120248189958140763509329170860, −8.675328453290467346412746162987, −7.56488205129248793878977963400, −6.82537028850410592495352329772, −6.06930706796753183656638355114, −4.80185692118510947708187108754, −3.94836177509535554787297865084, −3.05860759320748793684913141012, −1.61112609737087099095012156214, −0.24054109330313020973063016649,
1.68852746158432730657088559090, 2.75906206554837077468095360508, 3.68640283340405334095963975501, 4.97850359310795713734823341824, 5.91762054073863179527615287662, 6.35964449189387360158651058438, 7.954488516376281844351440766608, 8.200346659471150979457235400587, 9.311749527282980963037877267389, 9.720308818126509130726772446586
