L(s) = 1 | + (−0.178 − 1.72i)3-s + (−0.5 − 0.866i)5-s + (0.178 − 0.308i)7-s + (−2.93 + 0.614i)9-s + (−1.22 + 2.12i)11-s + (1.43 + 2.48i)13-s + (−1.40 + 1.01i)15-s + 0.872·17-s − 7.34·19-s + (−0.563 − 0.252i)21-s + (0.178 + 0.308i)23-s + (−0.499 + 0.866i)25-s + (1.58 + 4.94i)27-s + (−4.37 + 7.57i)29-s + (−4.38 − 7.59i)31-s + ⋯ |
L(s) = 1 | + (−0.102 − 0.994i)3-s + (−0.223 − 0.387i)5-s + (0.0673 − 0.116i)7-s + (−0.978 + 0.204i)9-s + (−0.369 + 0.639i)11-s + (0.398 + 0.690i)13-s + (−0.362 + 0.262i)15-s + 0.211·17-s − 1.68·19-s + (−0.122 − 0.0549i)21-s + (0.0371 + 0.0643i)23-s + (−0.0999 + 0.173i)25-s + (0.304 + 0.952i)27-s + (−0.812 + 1.40i)29-s + (−0.787 − 1.36i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0315 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0315 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4190259819\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4190259819\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.178 + 1.72i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
good | 7 | \( 1 + (-0.178 + 0.308i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.22 - 2.12i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.43 - 2.48i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 0.872T + 17T^{2} \) |
| 19 | \( 1 + 7.34T + 19T^{2} \) |
| 23 | \( 1 + (-0.178 - 0.308i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (4.37 - 7.57i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (4.38 + 7.59i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 6.87T + 37T^{2} \) |
| 41 | \( 1 + (-4.93 - 8.55i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.22 + 2.12i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.75 - 3.04i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 4T + 53T^{2} \) |
| 59 | \( 1 + (0.356 + 0.617i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.936 + 1.62i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.13 - 5.43i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 3.16T + 71T^{2} \) |
| 73 | \( 1 - 5.74T + 73T^{2} \) |
| 79 | \( 1 + (-6.32 + 10.9i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (6.50 - 11.2i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 1.25T + 89T^{2} \) |
| 97 | \( 1 + (6.87 - 11.9i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.541127817798450904399753536289, −8.809899040626298709550162087545, −8.051814647267908410450828663106, −7.32477560537067771117778152073, −6.57682330861082088206524158348, −5.73315027136404712680996856350, −4.74366241794064799909074620839, −3.75987814417553559238394910258, −2.35915732856958396086625064798, −1.46870259109114233442460401339,
0.16505397072170838678348247836, 2.30409148068877064700445689992, 3.41417029521922794731860479887, 4.04873735305136271954241857959, 5.21813736860699248451779392925, 5.85093625153744558959719460229, 6.76587340878673584536599890030, 7.974311673424769145362228759009, 8.556893889820362864801555045064, 9.301458155249739341323436004966