L(s) = 1 | + (0.450 + 1.34i)2-s + (−1.59 + 1.20i)4-s + (−0.739 + 1.78i)5-s + (0.385 − 0.385i)7-s + (−2.33 − 1.59i)8-s + (−2.72 − 0.188i)10-s + (−2.36 + 5.70i)11-s + (−2.30 + 0.956i)13-s + (0.690 + 0.343i)14-s + (1.08 − 3.84i)16-s + 5.05·17-s + (−1.27 − 3.07i)19-s + (−0.975 − 3.74i)20-s + (−8.71 − 0.601i)22-s + (−2.28 + 2.28i)23-s + ⋯ |
L(s) = 1 | + (0.318 + 0.948i)2-s + (−0.797 + 0.603i)4-s + (−0.330 + 0.798i)5-s + (0.145 − 0.145i)7-s + (−0.825 − 0.564i)8-s + (−0.862 − 0.0594i)10-s + (−0.712 + 1.72i)11-s + (−0.640 + 0.265i)13-s + (0.184 + 0.0917i)14-s + (0.271 − 0.962i)16-s + 1.22·17-s + (−0.292 − 0.705i)19-s + (−0.218 − 0.836i)20-s + (−1.85 − 0.128i)22-s + (−0.476 + 0.476i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.932 - 0.362i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.932 - 0.362i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.203734 + 1.08640i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.203734 + 1.08640i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.450 - 1.34i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.739 - 1.78i)T + (-3.53 - 3.53i)T^{2} \) |
| 7 | \( 1 + (-0.385 + 0.385i)T - 7iT^{2} \) |
| 11 | \( 1 + (2.36 - 5.70i)T + (-7.77 - 7.77i)T^{2} \) |
| 13 | \( 1 + (2.30 - 0.956i)T + (9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 - 5.05T + 17T^{2} \) |
| 19 | \( 1 + (1.27 + 3.07i)T + (-13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (2.28 - 2.28i)T - 23iT^{2} \) |
| 29 | \( 1 + (-0.735 + 0.304i)T + (20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + 3.40iT - 31T^{2} \) |
| 37 | \( 1 + (-9.56 - 3.96i)T + (26.1 + 26.1i)T^{2} \) |
| 41 | \( 1 + (-5.27 - 5.27i)T + 41iT^{2} \) |
| 43 | \( 1 + (-2.53 - 1.05i)T + (30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + 6.85iT - 47T^{2} \) |
| 53 | \( 1 + (7.45 + 3.08i)T + (37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (-6.14 - 2.54i)T + (41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (-2.67 - 6.46i)T + (-43.1 + 43.1i)T^{2} \) |
| 67 | \( 1 + (-10.2 + 4.26i)T + (47.3 - 47.3i)T^{2} \) |
| 71 | \( 1 + (6.37 + 6.37i)T + 71iT^{2} \) |
| 73 | \( 1 + (-9.03 + 9.03i)T - 73iT^{2} \) |
| 79 | \( 1 - 1.22T + 79T^{2} \) |
| 83 | \( 1 + (14.8 - 6.14i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (-4.97 + 4.97i)T - 89iT^{2} \) |
| 97 | \( 1 - 1.23T + 97T^{2} \) |
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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
−12.42109146275528499645453672340, −11.44050946425042115485491449995, −10.09253501169689342409830333444, −9.466152405964773617546777271420, −7.81885974268210111188097573632, −7.48045976338710060319255506386, −6.48868541361238852593875528852, −5.14795726058814651142285146711, −4.21861913193169311414754039332, −2.72968097495399409710569343000,
0.77515877707168938035519120482, 2.71776306898408550126441307277, 3.93485906776318247969677897348, 5.20010124395366631270231161517, 5.91667589597891311660855646299, 7.983489400333515736900483962053, 8.523990844549349984100716403755, 9.686466377243153274796870720293, 10.62099747250285303935189154610, 11.43907561817168244211229043024