L(s) = 1 | + (−0.988 − 0.149i)2-s + (0.955 + 0.294i)4-s + (0.0747 + 0.997i)5-s + (−0.900 + 0.433i)7-s + (−0.900 − 0.433i)8-s + (0.365 − 0.930i)9-s + (0.0747 − 0.997i)10-s + (0.0546 + 0.139i)11-s + (1.19 − 1.49i)13-s + (0.955 − 0.294i)14-s + (0.826 + 0.563i)16-s + (−0.5 + 0.866i)18-s + (−0.826 − 1.43i)19-s + (−0.222 + 0.974i)20-s + (−0.0332 − 0.145i)22-s + (0.733 − 0.680i)23-s + ⋯ |
L(s) = 1 | + (−0.988 − 0.149i)2-s + (0.955 + 0.294i)4-s + (0.0747 + 0.997i)5-s + (−0.900 + 0.433i)7-s + (−0.900 − 0.433i)8-s + (0.365 − 0.930i)9-s + (0.0747 − 0.997i)10-s + (0.0546 + 0.139i)11-s + (1.19 − 1.49i)13-s + (0.955 − 0.294i)14-s + (0.826 + 0.563i)16-s + (−0.5 + 0.866i)18-s + (−0.826 − 1.43i)19-s + (−0.222 + 0.974i)20-s + (−0.0332 − 0.145i)22-s + (0.733 − 0.680i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7148073005\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7148073005\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.988 + 0.149i)T \) |
| 5 | \( 1 + (-0.0747 - 0.997i)T \) |
| 7 | \( 1 + (0.900 - 0.433i)T \) |
good | 3 | \( 1 + (-0.365 + 0.930i)T^{2} \) |
| 11 | \( 1 + (-0.0546 - 0.139i)T + (-0.733 + 0.680i)T^{2} \) |
| 13 | \( 1 + (-1.19 + 1.49i)T + (-0.222 - 0.974i)T^{2} \) |
| 17 | \( 1 + (-0.0747 - 0.997i)T^{2} \) |
| 19 | \( 1 + (0.826 + 1.43i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.733 + 0.680i)T + (0.0747 - 0.997i)T^{2} \) |
| 29 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.698 + 0.215i)T + (0.826 - 0.563i)T^{2} \) |
| 41 | \( 1 + (-1.78 - 0.858i)T + (0.623 + 0.781i)T^{2} \) |
| 43 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 47 | \( 1 + (-1.44 - 0.218i)T + (0.955 + 0.294i)T^{2} \) |
| 53 | \( 1 + (1.40 + 0.432i)T + (0.826 + 0.563i)T^{2} \) |
| 59 | \( 1 + (0.134 - 1.79i)T + (-0.988 - 0.149i)T^{2} \) |
| 61 | \( 1 + (-0.826 + 0.563i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 73 | \( 1 + (-0.955 + 0.294i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 89 | \( 1 + (0.162 - 0.414i)T + (-0.733 - 0.680i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.278572481494481582095509012983, −8.772095100473543077271610451216, −7.76561434600274541644339272606, −6.95164831787249607897778291174, −6.31479060016187540064586023157, −5.85742537876488004184135632995, −4.04631791128703190294651952854, −3.03470261967907129963424429943, −2.62382599955642790408846671483, −0.833446909956757053208277165969,
1.23326125321139507692490907038, 2.09837681270143095606983851767, 3.64941525894431679923225346064, 4.46161278441061787531867943871, 5.77022271828080702816930389608, 6.28631226595229954083261151543, 7.25328609969530591446177260803, 7.967566779142487483219209573543, 8.762624680348547052668292711411, 9.319155863594858507240083576321