L(s) = 1 | + (0.766 − 0.642i)2-s + (0.0801 + 0.0291i)3-s + (0.173 − 0.984i)4-s + (0.0801 − 0.0291i)6-s + (−0.920 + 1.59i)7-s + (−0.500 − 0.866i)8-s + (−2.29 − 1.92i)9-s + (−1.21 − 2.09i)11-s + (0.0426 − 0.0738i)12-s + (−2.84 + 1.03i)13-s + (0.319 + 1.81i)14-s + (−0.939 − 0.342i)16-s + (5.22 − 4.38i)17-s − 2.99·18-s + (−3.15 − 3.00i)19-s + ⋯ |
L(s) = 1 | + (0.541 − 0.454i)2-s + (0.0462 + 0.0168i)3-s + (0.0868 − 0.492i)4-s + (0.0327 − 0.0119i)6-s + (−0.347 + 0.602i)7-s + (−0.176 − 0.306i)8-s + (−0.764 − 0.641i)9-s + (−0.364 − 0.632i)11-s + (0.0123 − 0.0213i)12-s + (−0.790 + 0.287i)13-s + (0.0854 + 0.484i)14-s + (−0.234 − 0.0855i)16-s + (1.26 − 1.06i)17-s − 0.705·18-s + (−0.724 − 0.689i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 + 0.581i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.813 + 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.385301 - 1.20102i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.385301 - 1.20102i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.766 + 0.642i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (3.15 + 3.00i)T \) |
good | 3 | \( 1 + (-0.0801 - 0.0291i)T + (2.29 + 1.92i)T^{2} \) |
| 7 | \( 1 + (0.920 - 1.59i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.21 + 2.09i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.84 - 1.03i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-5.22 + 4.38i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-1.39 + 7.90i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (7.77 + 6.52i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (2.41 - 4.18i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 0.202T + 37T^{2} \) |
| 41 | \( 1 + (-5.41 - 1.97i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.365 - 2.07i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-7.53 - 6.32i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-1.08 + 6.17i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (4.62 - 3.87i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-0.741 + 4.20i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-2.39 - 2.00i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (1.94 + 11.0i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-7.33 - 2.67i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (9.87 + 3.59i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (6.37 - 11.0i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (1.21 - 0.442i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (6.28 - 5.27i)T + (16.8 - 95.5i)T^{2} \) |
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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
−9.575480794281256020705921281159, −9.153485887004729230293097984301, −8.122003161918596844995652251296, −6.99453131584367961370593402667, −6.02082239394867636075864999896, −5.38229413426233088462716799269, −4.30350127679087236278527849895, −3.02839975044878009610470980625, −2.50126044686278924045995670533, −0.45740051293582233720356722030,
1.95647865080685587185735763650, 3.28889267036857752888572610968, 4.11237904121111709400528761072, 5.47120212436542712494548236079, 5.71156254625411242197676988047, 7.29269525271971314480888139380, 7.51529225435416938354769638090, 8.496819414998342845205800017261, 9.601980789681050309753593971364, 10.40063810479278034063297347044