| L(s) = 1 | + (0.998 − 0.0627i)2-s + (−0.904 + 0.425i)3-s + (0.992 − 0.125i)4-s + (−1.43 − 1.71i)5-s + (−0.876 + 0.481i)6-s + (1.49 + 2.06i)7-s + (0.982 − 0.187i)8-s + (0.637 − 0.770i)9-s + (−1.54 − 1.62i)10-s + (−0.289 − 4.60i)11-s + (−0.844 + 0.535i)12-s + (2.46 + 2.03i)13-s + (1.62 + 1.96i)14-s + (2.02 + 0.939i)15-s + (0.968 − 0.248i)16-s + (0.389 − 3.08i)17-s + ⋯ |
| L(s) = 1 | + (0.705 − 0.0443i)2-s + (−0.522 + 0.245i)3-s + (0.496 − 0.0626i)4-s + (−0.642 − 0.766i)5-s + (−0.357 + 0.196i)6-s + (0.566 + 0.780i)7-s + (0.347 − 0.0662i)8-s + (0.212 − 0.256i)9-s + (−0.487 − 0.512i)10-s + (−0.0873 − 1.38i)11-s + (−0.243 + 0.154i)12-s + (0.683 + 0.565i)13-s + (0.434 + 0.525i)14-s + (0.523 + 0.242i)15-s + (0.242 − 0.0621i)16-s + (0.0944 − 0.747i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.789 + 0.613i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.789 + 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.81718 - 0.622474i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.81718 - 0.622474i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.998 + 0.0627i)T \) |
| 3 | \( 1 + (0.904 - 0.425i)T \) |
| 5 | \( 1 + (1.43 + 1.71i)T \) |
| good | 7 | \( 1 + (-1.49 - 2.06i)T + (-2.16 + 6.65i)T^{2} \) |
| 11 | \( 1 + (0.289 + 4.60i)T + (-10.9 + 1.37i)T^{2} \) |
| 13 | \( 1 + (-2.46 - 2.03i)T + (2.43 + 12.7i)T^{2} \) |
| 17 | \( 1 + (-0.389 + 3.08i)T + (-16.4 - 4.22i)T^{2} \) |
| 19 | \( 1 + (-1.05 + 2.24i)T + (-12.1 - 14.6i)T^{2} \) |
| 23 | \( 1 + (-0.146 - 0.368i)T + (-16.7 + 15.7i)T^{2} \) |
| 29 | \( 1 + (-5.99 + 5.62i)T + (1.82 - 28.9i)T^{2} \) |
| 31 | \( 1 + (-8.06 - 1.01i)T + (30.0 + 7.70i)T^{2} \) |
| 37 | \( 1 + (1.74 + 6.77i)T + (-32.4 + 17.8i)T^{2} \) |
| 41 | \( 1 + (2.24 + 0.890i)T + (29.8 + 28.0i)T^{2} \) |
| 43 | \( 1 + (-0.383 - 0.124i)T + (34.7 + 25.2i)T^{2} \) |
| 47 | \( 1 + (-9.21 - 1.75i)T + (43.6 + 17.3i)T^{2} \) |
| 53 | \( 1 + (4.36 - 7.93i)T + (-28.3 - 44.7i)T^{2} \) |
| 59 | \( 1 + (3.29 + 5.19i)T + (-25.1 + 53.3i)T^{2} \) |
| 61 | \( 1 + (9.34 - 3.70i)T + (44.4 - 41.7i)T^{2} \) |
| 67 | \( 1 + (-0.826 + 0.879i)T + (-4.20 - 66.8i)T^{2} \) |
| 71 | \( 1 + (1.40 - 7.34i)T + (-66.0 - 26.1i)T^{2} \) |
| 73 | \( 1 + (4.07 + 2.58i)T + (31.0 + 66.0i)T^{2} \) |
| 79 | \( 1 + (2.40 + 5.10i)T + (-50.3 + 60.8i)T^{2} \) |
| 83 | \( 1 + (-7.44 - 3.50i)T + (52.9 + 63.9i)T^{2} \) |
| 89 | \( 1 + (9.37 - 14.7i)T + (-37.8 - 80.5i)T^{2} \) |
| 97 | \( 1 + (-2.14 - 2.28i)T + (-6.09 + 96.8i)T^{2} \) |
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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
−10.62819028822278695585876309354, −9.238044063535473891669129699965, −8.591055744637300211430357920895, −7.72307096065879654793995671178, −6.42813791020026639596999625905, −5.62805450115729998228233165328, −4.85568621377493953690741596867, −4.00723189669330822215805222735, −2.77987427264795441764339664493, −0.971955022152803245562467624099,
1.46420865362120872153184013967, 3.03148117506649114295011459902, 4.17278847525343350969464107085, 4.84352373700208428060391272625, 6.13309204587517173233749736393, 6.89189507515737765428602642823, 7.63046197083243067753137515235, 8.328795577286905366162405656433, 10.24476564920386235664634510643, 10.40745150006134055789188474556
