L(s) = 1 | + (−0.258 − 0.965i)2-s − i·3-s + (−0.866 + 0.499i)4-s + (0.939 − 3.50i)5-s + (−0.965 + 0.258i)6-s + (−2.04 + 1.67i)7-s + (0.707 + 0.707i)8-s − 9-s − 3.62·10-s + (−1.98 − 1.98i)11-s + (0.499 + 0.866i)12-s + (−3.16 + 1.71i)13-s + (2.15 + 1.53i)14-s + (−3.50 − 0.939i)15-s + (0.500 − 0.866i)16-s + (−1.98 − 3.44i)17-s + ⋯ |
L(s) = 1 | + (−0.183 − 0.683i)2-s − 0.577i·3-s + (−0.433 + 0.249i)4-s + (0.420 − 1.56i)5-s + (−0.394 + 0.105i)6-s + (−0.772 + 0.634i)7-s + (0.249 + 0.249i)8-s − 0.333·9-s − 1.14·10-s + (−0.598 − 0.598i)11-s + (0.144 + 0.249i)12-s + (−0.879 + 0.476i)13-s + (0.575 + 0.411i)14-s + (−0.905 − 0.242i)15-s + (0.125 − 0.216i)16-s + (−0.481 − 0.834i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.854 - 0.520i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.854 - 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.189145 + 0.674242i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.189145 + 0.674242i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.258 + 0.965i)T \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 + (2.04 - 1.67i)T \) |
| 13 | \( 1 + (3.16 - 1.71i)T \) |
good | 5 | \( 1 + (-0.939 + 3.50i)T + (-4.33 - 2.5i)T^{2} \) |
| 11 | \( 1 + (1.98 + 1.98i)T + 11iT^{2} \) |
| 17 | \( 1 + (1.98 + 3.44i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.23 - 1.23i)T + 19iT^{2} \) |
| 23 | \( 1 + (-2.41 - 1.39i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.41 + 4.19i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-9.26 + 2.48i)T + (26.8 - 15.5i)T^{2} \) |
| 37 | \( 1 + (11.2 - 3.00i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (2.09 - 7.80i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (1.95 + 1.12i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.00 - 0.538i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-2.21 + 3.83i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (5.27 + 1.41i)T + (51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + 13.6iT - 61T^{2} \) |
| 67 | \( 1 + (-2.58 + 2.58i)T - 67iT^{2} \) |
| 71 | \( 1 + (1.90 + 7.12i)T + (-61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (-1.41 - 5.26i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (2.35 + 4.08i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.55 - 4.55i)T + 83iT^{2} \) |
| 89 | \( 1 + (0.842 + 3.14i)T + (-77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-14.9 + 4.00i)T + (84.0 - 48.5i)T^{2} \) |
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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
−10.00302257387215751225173479426, −9.419271110183509497282178794385, −8.692619336104919611005596323946, −7.915088007906997505135540036564, −6.56444070497505885285269809073, −5.41907086970311761955289244165, −4.72623205049633483690604392504, −3.05689267109894013929318629718, −1.92968259683440699756416842854, −0.40760406778985767887232060971,
2.57014117752112451096682688530, 3.58538488397551987317943886158, 4.89715533436065906776227264613, 5.99792064738666914007348960421, 6.98261869412467380974707591031, 7.33206381966203512936921433407, 8.720974528770433907906511975449, 9.794536980627226941547222585803, 10.51314546229711101388930002732, 10.58655644654571748478681213522