L(s) = 1 | + (−0.342 + 0.593i)5-s + (1.80 − 1.93i)7-s + (−1.76 − 3.05i)11-s − 2.53·13-s + (−2.23 − 3.87i)17-s + (−1.46 + 2.52i)19-s + (1 − 1.73i)23-s + (2.26 + 3.92i)25-s − 6.52·29-s + (0.433 + 0.750i)31-s + (0.531 + 1.73i)35-s + (−1.26 + 2.19i)37-s − 11.6·41-s − 7.39·43-s + (−2.53 + 4.38i)47-s + ⋯ |
L(s) = 1 | + (−0.153 + 0.265i)5-s + (0.681 − 0.731i)7-s + (−0.532 − 0.922i)11-s − 0.702·13-s + (−0.542 − 0.939i)17-s + (−0.335 + 0.580i)19-s + (0.208 − 0.361i)23-s + (0.453 + 0.784i)25-s − 1.21·29-s + (0.0778 + 0.134i)31-s + (0.0897 + 0.292i)35-s + (−0.208 + 0.360i)37-s − 1.82·41-s − 1.12·43-s + (−0.369 + 0.639i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 + 0.353i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.935 + 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5280492080\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5280492080\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.80 + 1.93i)T \) |
good | 5 | \( 1 + (0.342 - 0.593i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.76 + 3.05i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 2.53T + 13T^{2} \) |
| 17 | \( 1 + (2.23 + 3.87i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.46 - 2.52i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1 + 1.73i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 6.52T + 29T^{2} \) |
| 31 | \( 1 + (-0.433 - 0.750i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.26 - 2.19i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 + 7.39T + 43T^{2} \) |
| 47 | \( 1 + (2.53 - 4.38i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4.12 + 7.15i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.76 - 8.25i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.53 + 4.38i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.82 - 4.90i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 9.06T + 71T^{2} \) |
| 73 | \( 1 + (-1.26 - 2.19i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.09 + 10.5i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 9.53T + 83T^{2} \) |
| 89 | \( 1 + (2.05 - 3.55i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 9.53T + 97T^{2} \) |
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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
−8.624025227638394486689681992842, −8.066019789364111837417294875017, −7.20305392162091897520271294159, −6.66626897078195272295347227223, −5.38776686186477411063249165691, −4.87247489080335147424109914220, −3.76857067811178888994515354793, −2.90920489733839715456695536755, −1.67504277183433012611701493647, −0.17541023076022122527814928693,
1.77107716770182797284447918480, 2.46555518251041896985139407049, 3.81873097474905170882839269418, 4.88627314653863865503377661745, 5.19723110367682868027411158413, 6.40623960103164455397230167384, 7.18599590120927683537860311412, 8.083092839892710559900274445089, 8.598104334166079464296806315240, 9.452863171490588679521803770252