L(s) = 1 | + (−1.59 + 0.672i)3-s + (4.18 + 0.738i)5-s + (1.26 + 1.51i)7-s + (2.09 − 2.14i)9-s + (−0.313 − 1.77i)11-s + (−3.16 + 1.15i)13-s + (−7.17 + 1.63i)15-s + (−1.51 − 0.873i)17-s + (3.62 − 2.09i)19-s + (−3.04 − 1.56i)21-s + (5.40 + 4.53i)23-s + (12.2 + 4.47i)25-s + (−1.90 + 4.83i)27-s + (−3.00 + 8.26i)29-s + (3.17 − 3.78i)31-s + ⋯ |
L(s) = 1 | + (−0.921 + 0.388i)3-s + (1.87 + 0.330i)5-s + (0.479 + 0.571i)7-s + (0.698 − 0.715i)9-s + (−0.0945 − 0.536i)11-s + (−0.878 + 0.319i)13-s + (−1.85 + 0.422i)15-s + (−0.366 − 0.211i)17-s + (0.832 − 0.480i)19-s + (−0.664 − 0.340i)21-s + (1.12 + 0.946i)23-s + (2.45 + 0.894i)25-s + (−0.366 + 0.930i)27-s + (−0.558 + 1.53i)29-s + (0.569 − 0.678i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.744 - 0.667i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.744 - 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.37015 + 0.523977i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.37015 + 0.523977i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.59 - 0.672i)T \) |
good | 5 | \( 1 + (-4.18 - 0.738i)T + (4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (-1.26 - 1.51i)T + (-1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (0.313 + 1.77i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (3.16 - 1.15i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (1.51 + 0.873i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.62 + 2.09i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.40 - 4.53i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (3.00 - 8.26i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-3.17 + 3.78i)T + (-5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (-0.864 + 1.49i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.58 + 4.34i)T + (-31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (9.92 - 1.74i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (2.63 - 2.20i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 - 0.511iT - 53T^{2} \) |
| 59 | \( 1 + (1.75 - 9.98i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-0.752 + 0.631i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (4.10 + 11.2i)T + (-51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-1.03 + 1.78i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.31 - 2.27i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.02 + 16.5i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-3.45 - 1.25i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (6.46 - 3.73i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.250 + 1.42i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−11.15069037382645044141741379689, −10.39086214265061525166813269723, −9.448738356573156956485406767018, −9.058354509156747454514625791428, −7.21882671812877477438786849493, −6.39010766529543474552921794178, −5.32897063722612185762602741739, −5.06752127975404585826035010112, −2.99725029280773316134367900886, −1.61494431747740078319311383829,
1.25243975592616411801694547564, 2.38491685579519587374679192105, 4.72107371591062388529110478085, 5.24854801094032674961640886097, 6.30544604737671193705083060930, 7.07031135884964308580062115511, 8.239896739201228920204177951869, 9.674767095058676730493014749497, 10.06276576918454604546507849398, 10.91825463310250156840187423797