| L(s) = 1 | + (−0.174 − 0.119i)2-s + (−0.714 − 1.82i)4-s + (3.75 − 1.15i)5-s + (1.85 − 1.88i)7-s + (−0.185 + 0.814i)8-s + (−0.792 − 0.244i)10-s + (0.0750 − 1.00i)11-s + (−2.55 + 1.23i)13-s + (−0.548 + 0.108i)14-s + (−2.73 + 2.53i)16-s + (−2.59 + 0.390i)17-s + (1.64 + 2.84i)19-s + (−4.78 − 6.00i)20-s + (−0.132 + 0.165i)22-s + (2.67 + 0.403i)23-s + ⋯ |
| L(s) = 1 | + (−0.123 − 0.0841i)2-s + (−0.357 − 0.910i)4-s + (1.67 − 0.517i)5-s + (0.700 − 0.713i)7-s + (−0.0657 + 0.288i)8-s + (−0.250 − 0.0772i)10-s + (0.0226 − 0.301i)11-s + (−0.708 + 0.341i)13-s + (−0.146 + 0.0290i)14-s + (−0.684 + 0.634i)16-s + (−0.628 + 0.0947i)17-s + (0.377 + 0.653i)19-s + (−1.06 − 1.34i)20-s + (−0.0281 + 0.0353i)22-s + (0.558 + 0.0841i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.250 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.250 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.27396 - 0.986357i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.27396 - 0.986357i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + (-1.85 + 1.88i)T \) |
| good | 2 | \( 1 + (0.174 + 0.119i)T + (0.730 + 1.86i)T^{2} \) |
| 5 | \( 1 + (-3.75 + 1.15i)T + (4.13 - 2.81i)T^{2} \) |
| 11 | \( 1 + (-0.0750 + 1.00i)T + (-10.8 - 1.63i)T^{2} \) |
| 13 | \( 1 + (2.55 - 1.23i)T + (8.10 - 10.1i)T^{2} \) |
| 17 | \( 1 + (2.59 - 0.390i)T + (16.2 - 5.01i)T^{2} \) |
| 19 | \( 1 + (-1.64 - 2.84i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.67 - 0.403i)T + (21.9 + 6.77i)T^{2} \) |
| 29 | \( 1 + (2.95 + 3.70i)T + (-6.45 + 28.2i)T^{2} \) |
| 31 | \( 1 + (1.33 - 2.31i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.02 + 2.60i)T + (-27.1 - 25.1i)T^{2} \) |
| 41 | \( 1 + (0.845 - 3.70i)T + (-36.9 - 17.7i)T^{2} \) |
| 43 | \( 1 + (0.320 + 1.40i)T + (-38.7 + 18.6i)T^{2} \) |
| 47 | \( 1 + (-4.46 - 3.04i)T + (17.1 + 43.7i)T^{2} \) |
| 53 | \( 1 + (-0.112 - 0.286i)T + (-38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (-7.68 - 2.37i)T + (48.7 + 33.2i)T^{2} \) |
| 61 | \( 1 + (-2.16 + 5.52i)T + (-44.7 - 41.4i)T^{2} \) |
| 67 | \( 1 + (5.38 - 9.33i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (5.08 - 6.37i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (10.5 - 7.22i)T + (26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 + (-2.63 - 4.56i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.55 - 0.748i)T + (51.7 + 64.8i)T^{2} \) |
| 89 | \( 1 + (1.00 + 13.4i)T + (-88.0 + 13.2i)T^{2} \) |
| 97 | \( 1 - 7.32T + 97T^{2} \) |
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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
−10.65426099135512747362424565722, −9.993808704861998229492392978225, −9.319304796400000604944211756765, −8.518295475392780520727947109004, −7.09727327313668601333693166891, −5.96960573485891008160198272228, −5.26950689978912492895815873623, −4.37759611594281622559672425627, −2.20044250574068949774195349349, −1.20233033304766493345308365378,
2.07983075823470177664337760770, 2.96072584663807801161389252013, 4.73831388274774157854902131885, 5.53963396419386684018561220310, 6.74738443038248738806247120007, 7.57979049124316987139868017386, 8.885751319375315819558393813610, 9.288823358958429066079522379523, 10.29759719291541964236891443489, 11.28987007631439364775227380967
