L(s) = 1 | + (1 + 1.73i)3-s + (−0.499 + 0.866i)9-s + (−2 − 3.46i)11-s + 4·13-s + (−1 − 1.73i)17-s + (3 − 5.19i)19-s + (4 − 6.92i)23-s + (2.5 + 4.33i)25-s + 4.00·27-s + 2·29-s + (2 + 3.46i)31-s + (3.99 − 6.92i)33-s + (−5 + 8.66i)37-s + (4 + 6.92i)39-s + 10·41-s + ⋯ |
L(s) = 1 | + (0.577 + 0.999i)3-s + (−0.166 + 0.288i)9-s + (−0.603 − 1.04i)11-s + 1.10·13-s + (−0.242 − 0.420i)17-s + (0.688 − 1.19i)19-s + (0.834 − 1.44i)23-s + (0.5 + 0.866i)25-s + 0.769·27-s + 0.371·29-s + (0.359 + 0.622i)31-s + (0.696 − 1.20i)33-s + (−0.821 + 1.42i)37-s + (0.640 + 1.10i)39-s + 1.56·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.196487341\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.196487341\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-1 - 1.73i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (2 + 3.46i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3 + 5.19i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4 + 6.92i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (5 - 8.66i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + (2 - 3.46i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1 - 1.73i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (5 + 8.66i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4 + 6.92i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (3 + 5.19i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (8 - 13.8i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 2T + 83T^{2} \) |
| 89 | \( 1 + (-9 + 15.5i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−9.246205449077501590188122054628, −8.819973047647685276071369880335, −8.209874330285570595765247364988, −7.01645411014812653776639112092, −6.22241341672656942791455858847, −5.06795835343994657195890386585, −4.49721812670018748879816353296, −3.17651384761456609021637342785, −2.96262410821103398149558108502, −0.956967959102428931238239117976,
1.25896718840293221643814796880, 2.12008480082472896061689294030, 3.20726947103440371066715122845, 4.26128872814453802511607629672, 5.40926782008359877060516321657, 6.27309481496124583338405877261, 7.24464315879427908437754913567, 7.70303083894424157131300750651, 8.445606938009543694679778250481, 9.247356259630485687235143637566