L(s) = 1 | + (0.268 − 1.38i)2-s + (1.63 − 0.577i)3-s + (−1.85 − 0.744i)4-s − 1.45i·5-s + (−0.364 − 2.42i)6-s + (−1.17 + 0.680i)7-s + (−1.53 + 2.37i)8-s + (2.33 − 1.88i)9-s + (−2.01 − 0.388i)10-s + (−0.711 + 1.23i)11-s + (−3.46 − 0.142i)12-s + (3.27 + 1.50i)13-s + (0.629 + 1.82i)14-s + (−0.838 − 2.36i)15-s + (2.89 + 2.76i)16-s + (1.68 − 0.971i)17-s + ⋯ |
L(s) = 1 | + (0.189 − 0.981i)2-s + (0.942 − 0.333i)3-s + (−0.928 − 0.372i)4-s − 0.648i·5-s + (−0.148 − 0.988i)6-s + (−0.445 + 0.257i)7-s + (−0.541 + 0.840i)8-s + (0.777 − 0.628i)9-s + (−0.637 − 0.122i)10-s + (−0.214 + 0.371i)11-s + (−0.999 − 0.0412i)12-s + (0.908 + 0.417i)13-s + (0.168 + 0.486i)14-s + (−0.216 − 0.611i)15-s + (0.723 + 0.690i)16-s + (0.408 − 0.235i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.215 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.215 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.900418 - 1.12039i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.900418 - 1.12039i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.268 + 1.38i)T \) |
| 3 | \( 1 + (-1.63 + 0.577i)T \) |
| 13 | \( 1 + (-3.27 - 1.50i)T \) |
good | 5 | \( 1 + 1.45iT - 5T^{2} \) |
| 7 | \( 1 + (1.17 - 0.680i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.711 - 1.23i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.68 + 0.971i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.01 - 1.16i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.96 - 3.39i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (7.48 + 4.32i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 6.55iT - 31T^{2} \) |
| 37 | \( 1 + (0.577 - 1.00i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.52 - 2.03i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-10.0 + 5.80i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 9.49T + 47T^{2} \) |
| 53 | \( 1 - 9.80iT - 53T^{2} \) |
| 59 | \( 1 + (-2.78 - 4.82i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.92 + 8.52i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (10.9 + 6.32i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (5.89 + 10.2i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 11.7T + 73T^{2} \) |
| 79 | \( 1 + 7.21iT - 79T^{2} \) |
| 83 | \( 1 - 0.869T + 83T^{2} \) |
| 89 | \( 1 + (-4.70 - 2.71i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5.22 + 9.05i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−12.71885417922517752177074331086, −11.95794943641513623951344598165, −10.60485455068876727800487407361, −9.413257873745090145516137735702, −8.897245628347501932084036869239, −7.72124188193622456230556213713, −5.99182256621982192276575541117, −4.40651534390353596932078771872, −3.20701839297878657801985375793, −1.64431048596525774752214117343,
3.10401902288404056638358140628, 4.13740398714130101034211350663, 5.79769984959323748999069779376, 6.97289086010954954195260609876, 7.990194051881563543684394312619, 8.886618666231609010602033719552, 9.971072682259200632159903141825, 10.95483851734684427061327599431, 12.89233230837405007147200194659, 13.31559705289449053403408137358