L(s) = 1 | + (0.597 + 0.802i)2-s + (−0.0581 + 0.998i)3-s + (−0.286 + 0.957i)4-s + (−0.835 + 0.549i)6-s + (−0.939 + 0.342i)8-s + (−0.993 − 0.116i)9-s + (−0.512 + 0.257i)11-s + (−0.939 − 0.342i)12-s + (−0.835 − 0.549i)16-s + (0.606 − 0.509i)17-s + (−0.499 − 0.866i)18-s + (1.36 + 1.14i)19-s + (−0.512 − 0.257i)22-s + (−0.286 − 0.957i)24-s + (0.396 − 0.918i)25-s + ⋯ |
L(s) = 1 | + (0.597 + 0.802i)2-s + (−0.0581 + 0.998i)3-s + (−0.286 + 0.957i)4-s + (−0.835 + 0.549i)6-s + (−0.939 + 0.342i)8-s + (−0.993 − 0.116i)9-s + (−0.512 + 0.257i)11-s + (−0.939 − 0.342i)12-s + (−0.835 − 0.549i)16-s + (0.606 − 0.509i)17-s + (−0.499 − 0.866i)18-s + (1.36 + 1.14i)19-s + (−0.512 − 0.257i)22-s + (−0.286 − 0.957i)24-s + (0.396 − 0.918i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.113447824\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.113447824\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.597 - 0.802i)T \) |
| 3 | \( 1 + (0.0581 - 0.998i)T \) |
good | 5 | \( 1 + (-0.396 + 0.918i)T^{2} \) |
| 7 | \( 1 + (0.0581 + 0.998i)T^{2} \) |
| 11 | \( 1 + (0.512 - 0.257i)T + (0.597 - 0.802i)T^{2} \) |
| 13 | \( 1 + (0.686 - 0.727i)T^{2} \) |
| 17 | \( 1 + (-0.606 + 0.509i)T + (0.173 - 0.984i)T^{2} \) |
| 19 | \( 1 + (-1.36 - 1.14i)T + (0.173 + 0.984i)T^{2} \) |
| 23 | \( 1 + (0.0581 - 0.998i)T^{2} \) |
| 29 | \( 1 + (-0.973 + 0.230i)T^{2} \) |
| 31 | \( 1 + (-0.893 + 0.448i)T^{2} \) |
| 37 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 41 | \( 1 + (0.0694 - 0.0932i)T + (-0.286 - 0.957i)T^{2} \) |
| 43 | \( 1 + (0.0694 - 1.19i)T + (-0.993 - 0.116i)T^{2} \) |
| 47 | \( 1 + (-0.893 - 0.448i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (1.49 + 0.749i)T + (0.597 + 0.802i)T^{2} \) |
| 61 | \( 1 + (0.835 - 0.549i)T^{2} \) |
| 67 | \( 1 + (-1.36 - 0.159i)T + (0.973 + 0.230i)T^{2} \) |
| 71 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 73 | \( 1 + (-1.86 + 0.679i)T + (0.766 - 0.642i)T^{2} \) |
| 79 | \( 1 + (0.286 - 0.957i)T^{2} \) |
| 83 | \( 1 + (-0.207 - 0.278i)T + (-0.286 + 0.957i)T^{2} \) |
| 89 | \( 1 + (1.43 - 0.524i)T + (0.766 - 0.642i)T^{2} \) |
| 97 | \( 1 + (1.62 + 1.06i)T + (0.396 + 0.918i)T^{2} \) |
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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
−11.18092270381328031280733654978, −10.02088608470243839351918782122, −9.465270638420113971307864587265, −8.301418561659949652843061119278, −7.69265198737071884622349342116, −6.46758723870284068637349936226, −5.47390610116312115380121974118, −4.87406046446822597067747504786, −3.75689535633358187668560010309, −2.86034020251550513474836739733,
1.18272167151859846625180239184, 2.57520447454442846389737597677, 3.47395866007221958681305328082, 5.05609490657813513837488820604, 5.69230048177721342257231823228, 6.78774046511568001612945096841, 7.69107837932426006034314667394, 8.800233988342310190503709205756, 9.645924482454335245697978543313, 10.79746293678126620137931071760