L(s) = 1 | + (1.19 − 0.755i)2-s + (0.881 + 0.471i)3-s + (0.857 − 1.80i)4-s + (0.324 − 0.396i)5-s + (1.41 − 0.103i)6-s + (−2.62 − 1.75i)7-s + (−0.341 − 2.80i)8-s + (0.555 + 0.831i)9-s + (0.0890 − 0.718i)10-s + (2.66 + 0.809i)11-s + (1.60 − 1.18i)12-s + (3.50 − 2.87i)13-s + (−4.46 − 0.112i)14-s + (0.473 − 0.196i)15-s + (−2.53 − 3.09i)16-s + (−1.24 − 0.514i)17-s + ⋯ |
L(s) = 1 | + (0.845 − 0.534i)2-s + (0.509 + 0.272i)3-s + (0.428 − 0.903i)4-s + (0.145 − 0.177i)5-s + (0.575 − 0.0421i)6-s + (−0.992 − 0.662i)7-s + (−0.120 − 0.992i)8-s + (0.185 + 0.277i)9-s + (0.0281 − 0.227i)10-s + (0.804 + 0.243i)11-s + (0.464 − 0.343i)12-s + (0.973 − 0.798i)13-s + (−1.19 − 0.0299i)14-s + (0.122 − 0.0506i)15-s + (−0.632 − 0.774i)16-s + (−0.301 − 0.124i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.446 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.446 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.08197 - 1.28836i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.08197 - 1.28836i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.19 + 0.755i)T \) |
| 3 | \( 1 + (-0.881 - 0.471i)T \) |
good | 5 | \( 1 + (-0.324 + 0.396i)T + (-0.975 - 4.90i)T^{2} \) |
| 7 | \( 1 + (2.62 + 1.75i)T + (2.67 + 6.46i)T^{2} \) |
| 11 | \( 1 + (-2.66 - 0.809i)T + (9.14 + 6.11i)T^{2} \) |
| 13 | \( 1 + (-3.50 + 2.87i)T + (2.53 - 12.7i)T^{2} \) |
| 17 | \( 1 + (1.24 + 0.514i)T + (12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (0.508 - 5.16i)T + (-18.6 - 3.70i)T^{2} \) |
| 23 | \( 1 + (-0.610 - 0.121i)T + (21.2 + 8.80i)T^{2} \) |
| 29 | \( 1 + (-1.53 - 5.06i)T + (-24.1 + 16.1i)T^{2} \) |
| 31 | \( 1 + (4.71 - 4.71i)T - 31iT^{2} \) |
| 37 | \( 1 + (0.337 - 0.0332i)T + (36.2 - 7.21i)T^{2} \) |
| 41 | \( 1 + (0.532 - 2.67i)T + (-37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (4.98 - 2.66i)T + (23.8 - 35.7i)T^{2} \) |
| 47 | \( 1 + (-2.42 + 5.85i)T + (-33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (-3.19 + 10.5i)T + (-44.0 - 29.4i)T^{2} \) |
| 59 | \( 1 + (10.5 + 8.66i)T + (11.5 + 57.8i)T^{2} \) |
| 61 | \( 1 + (1.48 - 2.77i)T + (-33.8 - 50.7i)T^{2} \) |
| 67 | \( 1 + (5.88 - 11.0i)T + (-37.2 - 55.7i)T^{2} \) |
| 71 | \( 1 + (4.98 - 7.45i)T + (-27.1 - 65.5i)T^{2} \) |
| 73 | \( 1 + (-6.36 + 4.25i)T + (27.9 - 67.4i)T^{2} \) |
| 79 | \( 1 + (-3.81 - 9.21i)T + (-55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (-12.4 - 1.22i)T + (81.4 + 16.1i)T^{2} \) |
| 89 | \( 1 + (-12.3 + 2.45i)T + (82.2 - 34.0i)T^{2} \) |
| 97 | \( 1 + (4.31 - 4.31i)T - 97iT^{2} \) |
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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
−11.09986252947441547157263552603, −10.32530705249393316551952674440, −9.613535169650799520918292629332, −8.635464121696420418617630915017, −7.14874762159738996609566663762, −6.29294771571079693766603621632, −5.14657035866900396714108742925, −3.76382713583253674623070498112, −3.31728821969332947459250254220, −1.47746986677270387049355725629,
2.31641331963299723725309191666, 3.42269140370008155292988479275, 4.46362214801544550403374400658, 6.22523988090503505899846601603, 6.33921029106982572069739512610, 7.54956972659997407885009421772, 8.882570928759017211725390472648, 9.171042501514572017787818352499, 10.81925962432706882078873813872, 11.80726465088447801809203936952