L(s) = 1 | + (0.5 − 0.866i)3-s + (1.5 + 2.59i)5-s + (0.5 + 2.59i)7-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)11-s + 4·13-s + 3·15-s + (−2 + 3.46i)17-s + (2.5 + 0.866i)21-s + (−4 − 6.92i)23-s + (−2 + 3.46i)25-s − 0.999·27-s + 7·29-s + (−5.5 + 9.52i)31-s + (0.499 + 0.866i)33-s + ⋯ |
L(s) = 1 | + (0.288 − 0.499i)3-s + (0.670 + 1.16i)5-s + (0.188 + 0.981i)7-s + (−0.166 − 0.288i)9-s + (−0.150 + 0.261i)11-s + 1.10·13-s + 0.774·15-s + (−0.485 + 0.840i)17-s + (0.545 + 0.188i)21-s + (−0.834 − 1.44i)23-s + (−0.400 + 0.692i)25-s − 0.192·27-s + 1.29·29-s + (−0.987 + 1.71i)31-s + (0.0870 + 0.150i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.032615435\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.032615435\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (-0.5 - 2.59i)T \) |
good | 5 | \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + (2 - 3.46i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4 + 6.92i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 7T + 29T^{2} \) |
| 31 | \( 1 + (5.5 - 9.52i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2 - 3.46i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 4T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 + (-1 - 1.73i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (5.5 - 9.52i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.5 + 6.06i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5 + 8.66i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + (-3 + 5.19i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.5 + 9.52i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 11T + 83T^{2} \) |
| 89 | \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 7T + 97T^{2} \) |
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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
−9.838693851739017473414669223754, −8.613086159089839350282324759769, −8.462824954838264001694209489614, −7.17635913379811346826055339069, −6.26865157089882679121800009773, −6.09231864265076290503379103225, −4.72869598950172916959730275789, −3.37357076077600121728009330204, −2.52015344641385632819445124923, −1.68529339660953502136328108613,
0.827964424476160638503126688935, 2.02653109127812968674929359939, 3.56082050619406454394368842853, 4.29059236587322857049971367076, 5.22079413497437144735511355551, 5.92284214698367475953356895183, 7.08683249791056411194129784235, 8.093482024827008278016674666494, 8.657346263488592570553212705801, 9.608586229737677162927271317162