L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.933 − 1.45i)3-s + (−0.499 + 0.866i)4-s + (0.230 − 0.398i)5-s + (−0.796 + 1.53i)6-s + 0.999·8-s + (−1.25 + 2.72i)9-s − 0.460·10-s + (1.82 + 3.15i)11-s + (1.73 − 0.0789i)12-s + (0.730 − 1.26i)13-s + (−0.796 + 0.0363i)15-s + (−0.5 − 0.866i)16-s + 3.73·17-s + (2.98 − 0.273i)18-s + 4.05·19-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.538 − 0.842i)3-s + (−0.249 + 0.433i)4-s + (0.102 − 0.178i)5-s + (−0.325 + 0.627i)6-s + 0.353·8-s + (−0.419 + 0.907i)9-s − 0.145·10-s + (0.549 + 0.952i)11-s + (0.499 − 0.0227i)12-s + (0.202 − 0.350i)13-s + (−0.205 + 0.00938i)15-s + (−0.125 − 0.216i)16-s + 0.905·17-s + (0.704 − 0.0643i)18-s + 0.930·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.262 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.262 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.890836 - 0.680821i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.890836 - 0.680821i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + (0.933 + 1.45i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.230 + 0.398i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.82 - 3.15i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.730 + 1.26i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 3.73T + 17T^{2} \) |
| 19 | \( 1 - 4.05T + 19T^{2} \) |
| 23 | \( 1 + (0.566 - 0.981i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.48 + 7.77i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.257 + 0.445i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 9.10T + 37T^{2} \) |
| 41 | \( 1 + (0.472 - 0.819i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.66 - 8.07i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1.16 + 2.01i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 12.4T + 53T^{2} \) |
| 59 | \( 1 + (-6.44 + 11.1i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (6.04 + 10.4i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.16 + 2.00i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 1.67T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 + (-2.50 - 4.33i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.32 - 5.75i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 2.72T + 89T^{2} \) |
| 97 | \( 1 + (5.59 + 9.68i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.725696836849615225642469037455, −9.520551768987174193318790931089, −8.001228170018251976496764261369, −7.67241881122921528429234182519, −6.58298625269844055386348498918, −5.61155205556350021531551825935, −4.64207091665540684871786927397, −3.30422655111653795607197022224, −1.99776644758136276598627397804, −0.953845517260355160793808502109,
0.980852798496423509232155223596, 3.13321588481853008287280101114, 4.12026957438879594242842655037, 5.26141050375358208064028375897, 5.94088050205427014328924901750, 6.72904050514977306337980017358, 7.79113434821116032042779985104, 8.862792826459418572161681297226, 9.346697080850185889166414250812, 10.27500624328303892981785981755