| L(s) = 1 | + (0.686 + 0.727i)2-s + (0.916 − 1.46i)3-s + (−0.0581 + 0.998i)4-s + (1.08 − 0.126i)5-s + (1.69 − 0.342i)6-s + (−0.387 − 0.194i)7-s + (−0.766 + 0.642i)8-s + (−1.32 − 2.69i)9-s + (0.834 + 0.700i)10-s + (0.774 + 1.79i)11-s + (1.41 + 1.00i)12-s + (−1.45 − 0.344i)13-s + (−0.124 − 0.415i)14-s + (0.806 − 1.70i)15-s + (−0.993 − 0.116i)16-s + (−0.868 + 4.92i)17-s + ⋯ |
| L(s) = 1 | + (0.485 + 0.514i)2-s + (0.529 − 0.848i)3-s + (−0.0290 + 0.499i)4-s + (0.484 − 0.0565i)5-s + (0.693 − 0.139i)6-s + (−0.146 − 0.0735i)7-s + (−0.270 + 0.227i)8-s + (−0.440 − 0.897i)9-s + (0.264 + 0.221i)10-s + (0.233 + 0.541i)11-s + (0.408 + 0.288i)12-s + (−0.402 − 0.0954i)13-s + (−0.0332 − 0.111i)14-s + (0.208 − 0.440i)15-s + (−0.248 − 0.0290i)16-s + (−0.210 + 1.19i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0584i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.67036 + 0.0488888i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.67036 + 0.0488888i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.686 - 0.727i)T \) |
| 3 | \( 1 + (-0.916 + 1.46i)T \) |
| good | 5 | \( 1 + (-1.08 + 0.126i)T + (4.86 - 1.15i)T^{2} \) |
| 7 | \( 1 + (0.387 + 0.194i)T + (4.18 + 5.61i)T^{2} \) |
| 11 | \( 1 + (-0.774 - 1.79i)T + (-7.54 + 8.00i)T^{2} \) |
| 13 | \( 1 + (1.45 + 0.344i)T + (11.6 + 5.83i)T^{2} \) |
| 17 | \( 1 + (0.868 - 4.92i)T + (-15.9 - 5.81i)T^{2} \) |
| 19 | \( 1 + (0.565 + 3.20i)T + (-17.8 + 6.49i)T^{2} \) |
| 23 | \( 1 + (-2.12 + 1.06i)T + (13.7 - 18.4i)T^{2} \) |
| 29 | \( 1 + (2.44 - 8.16i)T + (-24.2 - 15.9i)T^{2} \) |
| 31 | \( 1 + (5.16 + 3.39i)T + (12.2 + 28.4i)T^{2} \) |
| 37 | \( 1 + (-0.642 + 0.233i)T + (28.3 - 23.7i)T^{2} \) |
| 41 | \( 1 + (-2.80 + 2.97i)T + (-2.38 - 40.9i)T^{2} \) |
| 43 | \( 1 + (-3.51 + 4.72i)T + (-12.3 - 41.1i)T^{2} \) |
| 47 | \( 1 + (-2.10 + 1.38i)T + (18.6 - 43.1i)T^{2} \) |
| 53 | \( 1 + (1.14 - 1.97i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.55 + 10.5i)T + (-40.4 - 42.9i)T^{2} \) |
| 61 | \( 1 + (-0.817 - 14.0i)T + (-60.5 + 7.08i)T^{2} \) |
| 67 | \( 1 + (0.448 + 1.49i)T + (-55.9 + 36.8i)T^{2} \) |
| 71 | \( 1 + (-10.2 - 8.61i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-11.6 + 9.80i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-4.78 - 5.07i)T + (-4.59 + 78.8i)T^{2} \) |
| 83 | \( 1 + (-6.22 - 6.59i)T + (-4.82 + 82.8i)T^{2} \) |
| 89 | \( 1 + (8.27 - 6.94i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-14.2 - 1.66i)T + (94.3 + 22.3i)T^{2} \) |
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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
−12.90588849238887670342765114050, −12.43021806099303653974865071033, −11.04905727044433796881350440700, −9.542445657464545168572545772738, −8.624689197624595769645051200333, −7.42058907837422663754394913391, −6.62195896700173680759154627063, −5.44845059753908093286296374127, −3.77529388168501467580365562052, −2.12872958876282282120473697756,
2.40284642529944506437164152207, 3.68870471290341662728312462710, 4.96462170229026245793574890288, 6.09089332453955565267751578215, 7.78922238961288811056288225733, 9.242325779088626855347881944794, 9.739257667880089383119045111529, 10.87945826314113275415319958499, 11.72167768667008903742857394057, 13.03620353755002523053095995246
