L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.500 − 0.866i)4-s + (2 + 3.46i)5-s + (2 + 1.73i)7-s − 3·8-s + (1.99 − 3.46i)10-s + (−1 + 1.73i)11-s + 13-s + (0.499 − 2.59i)14-s + (0.500 + 0.866i)16-s + (3 − 5.19i)17-s + (−2 − 3.46i)19-s + 4·20-s + 1.99·22-s + (−3 − 5.19i)23-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.250 − 0.433i)4-s + (0.894 + 1.54i)5-s + (0.755 + 0.654i)7-s − 1.06·8-s + (0.632 − 1.09i)10-s + (−0.301 + 0.522i)11-s + 0.277·13-s + (0.133 − 0.694i)14-s + (0.125 + 0.216i)16-s + (0.727 − 1.26i)17-s + (−0.458 − 0.794i)19-s + 0.894·20-s + 0.426·22-s + (−0.625 − 1.08i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.20876 - 0.154037i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.20876 - 0.154037i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-2 - 1.73i)T \) |
good | 2 | \( 1 + (0.5 + 0.866i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (-2 - 3.46i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2 + 3.46i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + (1.5 - 2.59i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.5 + 2.59i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.5 - 4.33i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.5 - 6.06i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (-3 + 5.19i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.5 + 9.52i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + (-2 - 3.46i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 9T + 97T^{2} \) |
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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
−12.15426450409599511608426842844, −11.32705007789106230782260618041, −10.51661116892570110161586728721, −9.902330088778423068787203028341, −8.820695427120424241574013092030, −7.22406986068622315384115031300, −6.27570760253699139758996857516, −5.20866446729070686428160561998, −2.87627368937269559306127638353, −2.06956961744991906432622329790,
1.58396956880540069348832040896, 3.89375069261998848412767326432, 5.38175250394855804596159596545, 6.22333767532063700394517592893, 7.966233014049066905849841644330, 8.246535930921999168223185001496, 9.373838237260404718452307206033, 10.48024661562410809117275995165, 11.81025057505638265883708592369, 12.68522400706573042808288873370