| L(s) = 1 | + (0.587 − 0.809i)2-s + (0.951 − 0.309i)3-s + (−0.309 − 0.951i)4-s + (1.86 + 1.23i)5-s + (0.309 − 0.951i)6-s + 2.70i·7-s + (−0.951 − 0.309i)8-s + (0.809 − 0.587i)9-s + (2.09 − 0.786i)10-s + (−4.54 − 3.30i)11-s + (−0.587 − 0.809i)12-s + (−2.84 − 3.91i)13-s + (2.19 + 1.59i)14-s + (2.15 + 0.593i)15-s + (−0.809 + 0.587i)16-s + (0.994 + 0.323i)17-s + ⋯ |
| L(s) = 1 | + (0.415 − 0.572i)2-s + (0.549 − 0.178i)3-s + (−0.154 − 0.475i)4-s + (0.834 + 0.550i)5-s + (0.126 − 0.388i)6-s + 1.02i·7-s + (−0.336 − 0.109i)8-s + (0.269 − 0.195i)9-s + (0.661 − 0.248i)10-s + (−1.37 − 0.996i)11-s + (−0.169 − 0.233i)12-s + (−0.789 − 1.08i)13-s + (0.585 + 0.425i)14-s + (0.556 + 0.153i)15-s + (−0.202 + 0.146i)16-s + (0.241 + 0.0783i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 + 0.615i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.52551 - 0.525218i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.52551 - 0.525218i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.587 + 0.809i)T \) |
| 3 | \( 1 + (-0.951 + 0.309i)T \) |
| 5 | \( 1 + (-1.86 - 1.23i)T \) |
| good | 7 | \( 1 - 2.70iT - 7T^{2} \) |
| 11 | \( 1 + (4.54 + 3.30i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (2.84 + 3.91i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.994 - 0.323i)T + (13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (2.59 - 7.97i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-3.11 + 4.28i)T + (-7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (1.29 + 3.98i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (1.72 - 5.30i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-1.75 - 2.42i)T + (-11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-1.27 + 0.927i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 3.29iT - 43T^{2} \) |
| 47 | \( 1 + (-2.92 + 0.949i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-5.87 + 1.90i)T + (42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (11.4 - 8.33i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (0.218 + 0.159i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-6.00 - 1.94i)T + (54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (1.40 + 4.31i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-4.79 + 6.60i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (3.85 + 11.8i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-0.127 - 0.0415i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-9.53 - 6.93i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-2.51 + 0.815i)T + (78.4 - 57.0i)T^{2} \) |
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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
−12.88695961900462816433901218163, −12.22149115396931016591079474486, −10.63662698156745087546743999760, −10.17872685223726168796089363820, −8.844733151914900358329161221760, −7.81440162727950482049853341657, −6.04717352844010059578141010441, −5.32404186214776755132007573485, −3.15395405445264713373292220980, −2.32219329465399738568902724861,
2.42022892764829301230525190014, 4.40163314503455639244310349107, 5.15144116589814311900977525527, 6.91354326901691711319421766417, 7.61943890849607678080057446756, 9.091763664221481667283389961702, 9.805108586721541229147584507842, 11.01065644833086812424011744491, 12.68214395471470829740089472490, 13.25565817447270080944733859589
