L(s) = 1 | + (−1.40 − 0.192i)2-s − i·3-s + (1.92 + 0.540i)4-s + (−0.192 + 1.40i)6-s + 0.0802·7-s + (−2.59 − 1.12i)8-s − 9-s + 2.41i·11-s + (0.540 − 1.92i)12-s − 5.26i·13-s + (−0.112 − 0.0154i)14-s + (3.41 + 2.08i)16-s − 0.255·17-s + (1.40 + 0.192i)18-s − 6.95i·19-s + ⋯ |
L(s) = 1 | + (−0.990 − 0.136i)2-s − 0.577i·3-s + (0.962 + 0.270i)4-s + (−0.0786 + 0.571i)6-s + 0.0303·7-s + (−0.917 − 0.398i)8-s − 0.333·9-s + 0.728i·11-s + (0.155 − 0.555i)12-s − 1.46i·13-s + (−0.0300 − 0.00413i)14-s + (0.854 + 0.520i)16-s − 0.0620·17-s + (0.330 + 0.0454i)18-s − 1.59i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.398 + 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.398 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.414951 - 0.632939i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.414951 - 0.632939i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.40 + 0.192i)T \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 0.0802T + 7T^{2} \) |
| 11 | \( 1 - 2.41iT - 11T^{2} \) |
| 13 | \( 1 + 5.26iT - 13T^{2} \) |
| 17 | \( 1 + 0.255T + 17T^{2} \) |
| 19 | \( 1 + 6.95iT - 19T^{2} \) |
| 23 | \( 1 + 1.64T + 23T^{2} \) |
| 29 | \( 1 + 4.51iT - 29T^{2} \) |
| 31 | \( 1 - 8.29T + 31T^{2} \) |
| 37 | \( 1 + 2.67iT - 37T^{2} \) |
| 41 | \( 1 + 8.11T + 41T^{2} \) |
| 43 | \( 1 - 4.08iT - 43T^{2} \) |
| 47 | \( 1 - 5.70T + 47T^{2} \) |
| 53 | \( 1 + 11.5iT - 53T^{2} \) |
| 59 | \( 1 + 12.6iT - 59T^{2} \) |
| 61 | \( 1 + 11.9iT - 61T^{2} \) |
| 67 | \( 1 - 7.27iT - 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 - 5.50T + 79T^{2} \) |
| 83 | \( 1 - 9.20iT - 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 - 8.50T + 97T^{2} \) |
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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
−10.24852072879010873734932826543, −9.600196096228169872754610468794, −8.484288431818803722371873567362, −7.87720518140387954177910301964, −6.99787848386677411017290695292, −6.19014085155193549520018805891, −4.91736603252434577847005484074, −3.17849540150924894908872743328, −2.16003026334820548794991666551, −0.58512504985783757721729790981,
1.57981500712832667566358413793, 3.07763893670770847883304384191, 4.32785853268151873616162585883, 5.72931770482903544500466963067, 6.47742995602691348827216256658, 7.57059478624445808602453705256, 8.573087702279173983999306883351, 9.073393151139079912808529147499, 10.14787158926144051109923684199, 10.57196074994023466673123102393