L(s) = 1 | + (−0.854 − 0.854i)2-s − 0.539i·4-s + (−0.612 + 0.612i)5-s + (−2.64 − 0.0148i)7-s + (−2.17 + 2.17i)8-s + 1.04·10-s + (1.85 − 1.85i)11-s + (0.104 − 3.60i)13-s + (2.24 + 2.27i)14-s + 2.63·16-s − 3.04·17-s + (−0.104 + 0.104i)19-s + (0.330 + 0.330i)20-s − 3.17·22-s + 6.51i·23-s + ⋯ |
L(s) = 1 | + (−0.604 − 0.604i)2-s − 0.269i·4-s + (−0.274 + 0.274i)5-s + (−0.999 − 0.00559i)7-s + (−0.767 + 0.767i)8-s + 0.331·10-s + (0.559 − 0.559i)11-s + (0.0289 − 0.999i)13-s + (0.600 + 0.607i)14-s + 0.657·16-s − 0.739·17-s + (−0.0239 + 0.0239i)19-s + (0.0739 + 0.0739i)20-s − 0.675·22-s + 1.35i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.322 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.322 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.268909 + 0.192439i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.268909 + 0.192439i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (2.64 + 0.0148i)T \) |
| 13 | \( 1 + (-0.104 + 3.60i)T \) |
good | 2 | \( 1 + (0.854 + 0.854i)T + 2iT^{2} \) |
| 5 | \( 1 + (0.612 - 0.612i)T - 5iT^{2} \) |
| 11 | \( 1 + (-1.85 + 1.85i)T - 11iT^{2} \) |
| 17 | \( 1 + 3.04T + 17T^{2} \) |
| 19 | \( 1 + (0.104 - 0.104i)T - 19iT^{2} \) |
| 23 | \( 1 - 6.51iT - 23T^{2} \) |
| 29 | \( 1 - 3.78T + 29T^{2} \) |
| 31 | \( 1 + (6.77 - 6.77i)T - 31iT^{2} \) |
| 37 | \( 1 + (2.02 - 2.02i)T - 37iT^{2} \) |
| 41 | \( 1 + (2.27 - 2.27i)T - 41iT^{2} \) |
| 43 | \( 1 - 3.18iT - 43T^{2} \) |
| 47 | \( 1 + (-5.21 - 5.21i)T + 47iT^{2} \) |
| 53 | \( 1 + 3.43T + 53T^{2} \) |
| 59 | \( 1 + (9.15 + 9.15i)T + 59iT^{2} \) |
| 61 | \( 1 - 9.20iT - 61T^{2} \) |
| 67 | \( 1 + (1.04 + 1.04i)T + 67iT^{2} \) |
| 71 | \( 1 + (-4.10 - 4.10i)T + 71iT^{2} \) |
| 73 | \( 1 + (6.92 + 6.92i)T + 73iT^{2} \) |
| 79 | \( 1 - 17.5T + 79T^{2} \) |
| 83 | \( 1 + (10.5 - 10.5i)T - 83iT^{2} \) |
| 89 | \( 1 + (-3.39 - 3.39i)T + 89iT^{2} \) |
| 97 | \( 1 + (4.44 - 4.44i)T - 97iT^{2} \) |
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
−10.50859982226037227201417083128, −9.485878044732350046917888154372, −9.072744662993410598027568456878, −8.043173478347965591638303330800, −6.93366727163702990173037151156, −6.07033069348680107939469655549, −5.21444478846758212950939598100, −3.58005653063518753038450218227, −2.89640849305034891851334122362, −1.32430378115727671154825311617,
0.20900037132789734845765172446, 2.35449943855742302615763014396, 3.76083695191410496749275410629, 4.47404381123132102048240923216, 6.13554132718549763564672719352, 6.72761300299192542988024851505, 7.38927257347149023144767714872, 8.548083907271720342353620206898, 9.051623033584293257602905032977, 9.735273153055078008796454922531