| L(s) = 1 | + (0.978 − 0.207i)2-s + (0.0399 + 0.379i)3-s + (0.913 − 0.406i)4-s + (−0.669 + 0.743i)5-s + (0.118 + 0.363i)6-s + (−2.63 + 0.225i)7-s + (0.809 − 0.587i)8-s + (2.79 − 0.593i)9-s + (−0.5 + 0.866i)10-s + (3.19 + 0.907i)11-s + (0.190 + 0.330i)12-s + (−1.54 + 4.75i)13-s + (−2.53 + 0.768i)14-s + (−0.309 − 0.224i)15-s + (0.669 − 0.743i)16-s + (1.35 + 0.287i)17-s + ⋯ |
| L(s) = 1 | + (0.691 − 0.147i)2-s + (0.0230 + 0.219i)3-s + (0.456 − 0.203i)4-s + (−0.299 + 0.332i)5-s + (0.0481 + 0.148i)6-s + (−0.996 + 0.0853i)7-s + (0.286 − 0.207i)8-s + (0.930 − 0.197i)9-s + (−0.158 + 0.273i)10-s + (0.961 + 0.273i)11-s + (0.0551 + 0.0954i)12-s + (−0.428 + 1.31i)13-s + (−0.676 + 0.205i)14-s + (−0.0797 − 0.0579i)15-s + (0.167 − 0.185i)16-s + (0.327 + 0.0696i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.808 - 0.587i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.808 - 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.13426 + 0.693516i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.13426 + 0.693516i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.978 + 0.207i)T \) |
| 5 | \( 1 + (0.669 - 0.743i)T \) |
| 7 | \( 1 + (2.63 - 0.225i)T \) |
| 11 | \( 1 + (-3.19 - 0.907i)T \) |
| good | 3 | \( 1 + (-0.0399 - 0.379i)T + (-2.93 + 0.623i)T^{2} \) |
| 13 | \( 1 + (1.54 - 4.75i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-1.35 - 0.287i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (-6.39 - 2.84i)T + (12.7 + 14.1i)T^{2} \) |
| 23 | \( 1 + (0.927 + 1.60i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.190 - 0.138i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-4.42 - 4.91i)T + (-3.24 + 30.8i)T^{2} \) |
| 37 | \( 1 + (-0.362 + 3.45i)T + (-36.1 - 7.69i)T^{2} \) |
| 41 | \( 1 + (-2.5 + 1.81i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 9.70T + 43T^{2} \) |
| 47 | \( 1 + (4.86 + 2.16i)T + (31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (-5.31 - 5.90i)T + (-5.54 + 52.7i)T^{2} \) |
| 59 | \( 1 + (12.0 - 5.36i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (-2.83 + 3.14i)T + (-6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (-4.59 + 7.95i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (2.20 + 6.79i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-3.38 + 1.50i)T + (48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (15.9 - 3.39i)T + (72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (1.69 + 5.20i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (8.97 + 15.5i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.01 - 3.13i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.21727624553389777991957189808, −9.776244996479741516695805981693, −8.962199120689956445663602608731, −7.42131096042587029751928796331, −6.84725826521973339440793457630, −6.10889730204853758508674153442, −4.75783522892981207497211580117, −3.91692084005306165301899028874, −3.14741348213875250645900754312, −1.58185211095032997977365364902,
1.04599791624139187913089583826, 2.88896899847007200052025211651, 3.72029081255057368913043989703, 4.81111454304857794273837255560, 5.77245996417072786420692296169, 6.76022829617897775527463029167, 7.44761270894987047297449928541, 8.312728126997965934558922885455, 9.684548692947925200615786153060, 9.986668591476806906602398542930
