L(s) = 1 | + (1.23 + 2.13i)2-s + (−0.933 − 1.45i)3-s + (−2.02 + 3.51i)4-s + 2.59·5-s + (1.96 − 3.78i)6-s − 5.05·8-s + (−1.25 + 2.72i)9-s + (3.19 + 5.52i)10-s + 4.51·11-s + (7.01 − 0.319i)12-s + (0.5 + 0.866i)13-s + (−2.42 − 3.78i)15-s + (−2.16 − 3.74i)16-s + (−0.472 − 0.819i)17-s + (−7.35 + 0.671i)18-s + (−2.02 + 3.51i)19-s + ⋯ |
L(s) = 1 | + (0.869 + 1.50i)2-s + (−0.538 − 0.842i)3-s + (−1.01 + 1.75i)4-s + 1.15·5-s + (0.800 − 1.54i)6-s − 1.78·8-s + (−0.419 + 0.907i)9-s + (1.00 + 1.74i)10-s + 1.36·11-s + (2.02 − 0.0923i)12-s + (0.138 + 0.240i)13-s + (−0.625 − 0.977i)15-s + (−0.540 − 0.936i)16-s + (−0.114 − 0.198i)17-s + (−1.73 + 0.158i)18-s + (−0.465 + 0.805i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.37146 + 1.62713i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.37146 + 1.62713i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.933 + 1.45i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-1.23 - 2.13i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 - 2.59T + 5T^{2} \) |
| 11 | \( 1 - 4.51T + 11T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (0.472 + 0.819i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.02 - 3.51i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 0.273T + 23T^{2} \) |
| 29 | \( 1 + (1.23 - 2.13i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.16 + 2.01i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.890 - 1.54i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3.20 + 5.54i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.21 + 9.03i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (6.08 + 10.5i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.13 - 5.43i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.36 - 2.36i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.13 + 1.96i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.90 + 13.6i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 3.27T + 71T^{2} \) |
| 73 | \( 1 + (0.753 + 1.30i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (7.35 + 12.7i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.472 - 0.819i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (7.17 - 12.4i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (5.74 - 9.95i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.83080069123767113628684616197, −10.50093852559701872107002927203, −9.212505378835656785435703411763, −8.348043107380856614555009895675, −7.19902525062004656661259970730, −6.51434155236879232154329866942, −5.93349882772491475329813245730, −5.13130296208713931076443849505, −3.84545812492656520123080674831, −1.85291228192384056297959678623,
1.33369048171989525893683472762, 2.76001266267839688184204914914, 3.94948027167378425628095772862, 4.76563959683769665040461769596, 5.79583286830482968443290047630, 6.50400479468265421627945579070, 8.811234606456644798036577199631, 9.661036690781083807922293437987, 10.03479940266959488818776922385, 11.16041573266512027779077693702