| L(s) = 1 | + (−0.952 − 1.04i)2-s + (−1.51 − 1.82i)3-s + (−0.183 + 1.99i)4-s + (−1.84 + 1.26i)5-s + (−0.469 + 3.32i)6-s + (1.14 − 0.373i)7-s + (2.25 − 1.70i)8-s + (−0.493 + 2.58i)9-s + (3.07 + 0.717i)10-s + (0.0547 − 0.433i)11-s + (3.92 − 2.67i)12-s + (1.06 − 5.56i)13-s + (−1.48 − 0.844i)14-s + (5.10 + 1.45i)15-s + (−3.93 − 0.732i)16-s + (−0.908 − 3.54i)17-s + ⋯ |
| L(s) = 1 | + (−0.673 − 0.738i)2-s + (−0.873 − 1.05i)3-s + (−0.0918 + 0.995i)4-s + (−0.823 + 0.566i)5-s + (−0.191 + 1.35i)6-s + (0.434 − 0.141i)7-s + (0.797 − 0.603i)8-s + (−0.164 + 0.862i)9-s + (0.973 + 0.226i)10-s + (0.0164 − 0.130i)11-s + (1.13 − 0.772i)12-s + (0.294 − 1.54i)13-s + (−0.396 − 0.225i)14-s + (1.31 + 0.374i)15-s + (−0.983 − 0.183i)16-s + (−0.220 − 0.858i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.104 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.104 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0889247 + 0.0988028i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0889247 + 0.0988028i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.952 + 1.04i)T \) |
| 5 | \( 1 + (1.84 - 1.26i)T \) |
| good | 3 | \( 1 + (1.51 + 1.82i)T + (-0.562 + 2.94i)T^{2} \) |
| 7 | \( 1 + (-1.14 + 0.373i)T + (5.66 - 4.11i)T^{2} \) |
| 11 | \( 1 + (-0.0547 + 0.433i)T + (-10.6 - 2.73i)T^{2} \) |
| 13 | \( 1 + (-1.06 + 5.56i)T + (-12.0 - 4.78i)T^{2} \) |
| 17 | \( 1 + (0.908 + 3.54i)T + (-14.8 + 8.18i)T^{2} \) |
| 19 | \( 1 + (0.713 + 0.590i)T + (3.56 + 18.6i)T^{2} \) |
| 23 | \( 1 + (2.46 - 2.62i)T + (-1.44 - 22.9i)T^{2} \) |
| 29 | \( 1 + (1.13 - 0.0714i)T + (28.7 - 3.63i)T^{2} \) |
| 31 | \( 1 + (1.91 - 0.492i)T + (27.1 - 14.9i)T^{2} \) |
| 37 | \( 1 + (-5.46 - 3.00i)T + (19.8 + 31.2i)T^{2} \) |
| 41 | \( 1 + (1.01 - 0.953i)T + (2.57 - 40.9i)T^{2} \) |
| 43 | \( 1 + (0.917 - 0.666i)T + (13.2 - 40.8i)T^{2} \) |
| 47 | \( 1 + (-0.163 - 0.414i)T + (-34.2 + 32.1i)T^{2} \) |
| 53 | \( 1 + (-3.35 + 5.28i)T + (-22.5 - 47.9i)T^{2} \) |
| 59 | \( 1 + (11.8 - 5.58i)T + (37.6 - 45.4i)T^{2} \) |
| 61 | \( 1 + (-5.48 + 5.84i)T + (-3.83 - 60.8i)T^{2} \) |
| 67 | \( 1 + (-0.157 + 2.50i)T + (-66.4 - 8.39i)T^{2} \) |
| 71 | \( 1 + (-1.62 + 0.643i)T + (51.7 - 48.6i)T^{2} \) |
| 73 | \( 1 + (11.4 + 5.37i)T + (46.5 + 56.2i)T^{2} \) |
| 79 | \( 1 + (0.162 + 0.196i)T + (-14.8 + 77.6i)T^{2} \) |
| 83 | \( 1 + (4.26 - 5.15i)T + (-15.5 - 81.5i)T^{2} \) |
| 89 | \( 1 + (4.54 - 9.65i)T + (-56.7 - 68.5i)T^{2} \) |
| 97 | \( 1 + (-7.93 + 0.499i)T + (96.2 - 12.1i)T^{2} \) |
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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
−9.492443909991096868245480922382, −8.225059611976791292659781538731, −7.76202053378606069791388546464, −7.10094819439435319210814292347, −6.16459314325340137080700521429, −4.95975962831310569299393904318, −3.66793022063997875918495489194, −2.65538421219195759031946409169, −1.19495991440848228813316401568, −0.098775337807588468148548527664,
1.69884571223267621279671883228, 4.16528947176552493421226766865, 4.41432919897567228862862186243, 5.46307110167803932111663443200, 6.25833770010383475756471578479, 7.26321591928912268453175158431, 8.261301710357903832488593101260, 8.898112386374693362391246534175, 9.658298303310616702350920825854, 10.52953018907459576231164876934
