| L(s) = 1 | + (0.308 + 0.0731i)2-s + (−0.589 − 1.62i)3-s + (−1.69 − 0.852i)4-s + (1.26 − 1.70i)5-s + (−0.0626 − 0.545i)6-s + (2.12 + 1.39i)7-s + (−0.947 − 0.795i)8-s + (−2.30 + 1.91i)9-s + (0.515 − 0.432i)10-s + (3.03 − 0.355i)11-s + (−0.388 + 3.26i)12-s + (−1.58 + 5.28i)13-s + (0.553 + 0.586i)14-s + (−3.51 − 1.06i)15-s + (2.03 + 2.73i)16-s + (−0.745 − 4.22i)17-s + ⋯ |
| L(s) = 1 | + (0.218 + 0.0517i)2-s + (−0.340 − 0.940i)3-s + (−0.848 − 0.426i)4-s + (0.566 − 0.760i)5-s + (−0.0255 − 0.222i)6-s + (0.803 + 0.528i)7-s + (−0.335 − 0.281i)8-s + (−0.768 + 0.639i)9-s + (0.163 − 0.136i)10-s + (0.916 − 0.107i)11-s + (−0.112 + 0.943i)12-s + (−0.439 + 1.46i)13-s + (0.148 + 0.156i)14-s + (−0.908 − 0.273i)15-s + (0.508 + 0.683i)16-s + (−0.180 − 1.02i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.423 + 0.905i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.423 + 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.777730 - 0.494635i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.777730 - 0.494635i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.589 + 1.62i)T \) |
| good | 2 | \( 1 + (-0.308 - 0.0731i)T + (1.78 + 0.897i)T^{2} \) |
| 5 | \( 1 + (-1.26 + 1.70i)T + (-1.43 - 4.78i)T^{2} \) |
| 7 | \( 1 + (-2.12 - 1.39i)T + (2.77 + 6.42i)T^{2} \) |
| 11 | \( 1 + (-3.03 + 0.355i)T + (10.7 - 2.53i)T^{2} \) |
| 13 | \( 1 + (1.58 - 5.28i)T + (-10.8 - 7.14i)T^{2} \) |
| 17 | \( 1 + (0.745 + 4.22i)T + (-15.9 + 5.81i)T^{2} \) |
| 19 | \( 1 + (0.0105 - 0.0600i)T + (-17.8 - 6.49i)T^{2} \) |
| 23 | \( 1 + (-3.57 + 2.35i)T + (9.10 - 21.1i)T^{2} \) |
| 29 | \( 1 + (7.32 - 7.76i)T + (-1.68 - 28.9i)T^{2} \) |
| 31 | \( 1 + (0.268 + 4.60i)T + (-30.7 + 3.59i)T^{2} \) |
| 37 | \( 1 + (3.30 + 1.20i)T + (28.3 + 23.7i)T^{2} \) |
| 41 | \( 1 + (-2.45 + 0.581i)T + (36.6 - 18.4i)T^{2} \) |
| 43 | \( 1 + (0.307 - 0.712i)T + (-29.5 - 31.2i)T^{2} \) |
| 47 | \( 1 + (-0.264 + 4.53i)T + (-46.6 - 5.45i)T^{2} \) |
| 53 | \( 1 + (-0.986 - 1.70i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (7.16 + 0.837i)T + (57.4 + 13.6i)T^{2} \) |
| 61 | \( 1 + (10.4 - 5.24i)T + (36.4 - 48.9i)T^{2} \) |
| 67 | \( 1 + (-0.832 - 0.882i)T + (-3.89 + 66.8i)T^{2} \) |
| 71 | \( 1 + (10.9 - 9.17i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-7.44 - 6.24i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-1.56 - 0.371i)T + (70.5 + 35.4i)T^{2} \) |
| 83 | \( 1 + (-3.21 - 0.761i)T + (74.1 + 37.2i)T^{2} \) |
| 89 | \( 1 + (10.7 + 9.02i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-4.19 - 5.63i)T + (-27.8 + 92.9i)T^{2} \) |
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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
−14.08269415226261734884288618699, −13.17581155102645387448111051662, −12.16875609015996899265735669380, −11.23807449084531476227291360621, −9.258295376219912450104092900730, −8.825000412295572690613597647570, −7.02995415378625115970111474754, −5.62216061088314051019989787057, −4.70312118537462558279286073955, −1.61333590746963738533378648687,
3.41707088627495908795218529587, 4.69122884836001320207505307882, 5.96946684498544380402042530767, 7.81907219794170290569102040392, 9.166006465112173541931057777381, 10.23415857995206543952954160423, 11.09621980656064539392131110359, 12.41040712998897739146136747561, 13.70164916693880423770251924148, 14.62508683844485399898194511735
