| L(s) = 1 | + (−0.309 − 0.951i)2-s + (0.978 − 0.207i)3-s + (−0.809 + 0.587i)4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + (2.35 + 1.05i)7-s + (0.809 + 0.587i)8-s + (0.913 − 0.406i)9-s + (0.978 + 0.207i)10-s + (0.100 + 0.959i)11-s + (−0.669 + 0.743i)12-s + (−2.36 − 2.63i)13-s + (0.269 − 2.56i)14-s + (−0.309 + 0.951i)15-s + (0.309 − 0.951i)16-s + (0.00862 − 0.0820i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 − 0.672i)2-s + (0.564 − 0.120i)3-s + (−0.404 + 0.293i)4-s + (−0.223 + 0.387i)5-s + (−0.204 − 0.353i)6-s + (0.891 + 0.396i)7-s + (0.286 + 0.207i)8-s + (0.304 − 0.135i)9-s + (0.309 + 0.0657i)10-s + (0.0304 + 0.289i)11-s + (−0.193 + 0.214i)12-s + (−0.657 − 0.729i)13-s + (0.0721 − 0.686i)14-s + (−0.0797 + 0.245i)15-s + (0.0772 − 0.237i)16-s + (0.00209 − 0.0199i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.373i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 + 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.73854 - 0.336668i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.73854 - 0.336668i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.309 + 0.951i)T \) |
| 3 | \( 1 + (-0.978 + 0.207i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 + (-5.16 - 2.08i)T \) |
| good | 7 | \( 1 + (-2.35 - 1.05i)T + (4.68 + 5.20i)T^{2} \) |
| 11 | \( 1 + (-0.100 - 0.959i)T + (-10.7 + 2.28i)T^{2} \) |
| 13 | \( 1 + (2.36 + 2.63i)T + (-1.35 + 12.9i)T^{2} \) |
| 17 | \( 1 + (-0.00862 + 0.0820i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (-2.41 + 2.67i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (-5.87 - 4.26i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-3.03 - 9.34i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (-2.85 - 4.94i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.92 + 0.408i)T + (37.4 + 16.6i)T^{2} \) |
| 43 | \( 1 + (-8.17 + 9.07i)T + (-4.49 - 42.7i)T^{2} \) |
| 47 | \( 1 + (0.396 - 1.22i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-7.49 + 3.33i)T + (35.4 - 39.3i)T^{2} \) |
| 59 | \( 1 + (-6.15 + 1.30i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + 9.67T + 61T^{2} \) |
| 67 | \( 1 + (6.06 - 10.5i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (1.39 - 0.621i)T + (47.5 - 52.7i)T^{2} \) |
| 73 | \( 1 + (0.918 + 8.73i)T + (-71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (-1.30 + 12.4i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (5.88 + 1.25i)T + (75.8 + 33.7i)T^{2} \) |
| 89 | \( 1 + (7.31 - 5.31i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-9.70 + 7.05i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.10445301786608702384970938323, −9.112593866775146175802965656646, −8.512498025739278350252358116461, −7.56884307581408140430402347325, −6.99155643706863103670049831019, −5.35577305088219459766559801304, −4.65444902450650871875243219310, −3.28401745138298678419068904956, −2.58546420739937113117374541909, −1.26865782240284307247129892597,
1.06955566979462262737744633043, 2.59569078914326369698571629979, 4.21304872320872274430664519470, 4.64137759207384221590102155295, 5.82206920474423860380619739890, 6.92622863816810399341572283126, 7.79352274584467052259573235628, 8.252476853275353367933748418309, 9.183152928232053169299264209784, 9.835536943681632252275332915337
