L(s) = 1 | + (0.174 − 1.40i)2-s + (0.618 − 1.61i)3-s + (−1.93 − 0.489i)4-s + (−2.46 + 1.42i)5-s + (−2.16 − 1.14i)6-s + (−1.02 − 2.44i)7-s + (−1.02 + 2.63i)8-s + (−2.23 − 2.00i)9-s + (1.57 + 3.71i)10-s + (2.42 − 4.20i)11-s + (−1.99 + 2.83i)12-s + 2.75·13-s + (−3.60 + 1.00i)14-s + (0.780 + 4.87i)15-s + (3.52 + 1.89i)16-s + (1.75 − 3.03i)17-s + ⋯ |
L(s) = 1 | + (0.123 − 0.992i)2-s + (0.356 − 0.934i)3-s + (−0.969 − 0.244i)4-s + (−1.10 + 0.637i)5-s + (−0.883 − 0.469i)6-s + (−0.385 − 0.922i)7-s + (−0.362 + 0.931i)8-s + (−0.745 − 0.666i)9-s + (0.496 + 1.17i)10-s + (0.731 − 1.26i)11-s + (−0.574 + 0.818i)12-s + 0.763·13-s + (−0.963 + 0.268i)14-s + (0.201 + 1.25i)15-s + (0.880 + 0.474i)16-s + (0.425 − 0.736i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 + 0.305i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.952 + 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.148850 - 0.951457i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.148850 - 0.951457i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.174 + 1.40i)T \) |
| 3 | \( 1 + (-0.618 + 1.61i)T \) |
| 7 | \( 1 + (1.02 + 2.44i)T \) |
good | 5 | \( 1 + (2.46 - 1.42i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.42 + 4.20i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 2.75T + 13T^{2} \) |
| 17 | \( 1 + (-1.75 + 3.03i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.14 - 5.44i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.15 + 1.82i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 3.90T + 29T^{2} \) |
| 31 | \( 1 + (0.858 + 0.495i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.06 + 0.614i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 2.10T + 41T^{2} \) |
| 43 | \( 1 + 5.11iT - 43T^{2} \) |
| 47 | \( 1 + (-5.61 - 9.72i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.00 + 1.73i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.890 - 0.514i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.24 - 2.15i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.02 - 2.90i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 9.75iT - 71T^{2} \) |
| 73 | \( 1 + (-0.291 - 0.168i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.80 + 4.85i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 0.138iT - 83T^{2} \) |
| 89 | \( 1 + (-0.580 - 1.00i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 11.0iT - 97T^{2} \) |
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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
−12.18708756167874244863949877573, −11.43840532797571345960777379855, −10.75260461114004296440193362603, −9.313204537821797757425764042079, −8.207112539751032438850194403352, −7.28686579972778912206869648497, −5.96732781678518758617244252223, −3.74835735158957128701525852745, −3.25579334641627327916952058237, −0.932425899737755812913475970357,
3.51812770089477855226107331126, 4.50329988885137505341073685553, 5.53549813715434181180735773745, 7.07312742223068903209976252968, 8.277094138072261691484357728229, 9.012555758003547177491952657521, 9.736037450187798928979178102674, 11.41571871617959118607790724170, 12.36779620266598931176701661863, 13.31553701967501104817411362435