| L(s) = 1 | + (−0.809 + 0.587i)2-s + (−0.913 + 0.406i)3-s + (0.309 − 0.951i)4-s + (0.5 + 0.866i)5-s + (0.5 − 0.866i)6-s + (2.57 + 2.86i)7-s + (0.309 + 0.951i)8-s + (0.669 − 0.743i)9-s + (−0.913 − 0.406i)10-s + (−6.39 + 1.35i)11-s + (0.104 + 0.994i)12-s + (0.158 − 1.51i)13-s + (−3.77 − 0.801i)14-s + (−0.809 − 0.587i)15-s + (−0.809 − 0.587i)16-s + (−0.709 − 0.150i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 + 0.415i)2-s + (−0.527 + 0.234i)3-s + (0.154 − 0.475i)4-s + (0.223 + 0.387i)5-s + (0.204 − 0.353i)6-s + (0.975 + 1.08i)7-s + (0.109 + 0.336i)8-s + (0.223 − 0.247i)9-s + (−0.288 − 0.128i)10-s + (−1.92 + 0.409i)11-s + (0.0301 + 0.287i)12-s + (0.0440 − 0.418i)13-s + (−1.00 − 0.214i)14-s + (−0.208 − 0.151i)15-s + (−0.202 − 0.146i)16-s + (−0.171 − 0.0365i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00130679 + 0.588525i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00130679 + 0.588525i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.809 - 0.587i)T \) |
| 3 | \( 1 + (0.913 - 0.406i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 31 | \( 1 + (0.890 - 5.49i)T \) |
| good | 7 | \( 1 + (-2.57 - 2.86i)T + (-0.731 + 6.96i)T^{2} \) |
| 11 | \( 1 + (6.39 - 1.35i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (-0.158 + 1.51i)T + (-12.7 - 2.70i)T^{2} \) |
| 17 | \( 1 + (0.709 + 0.150i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (-0.530 - 5.05i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (0.185 + 0.571i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-2.85 + 2.07i)T + (8.96 - 27.5i)T^{2} \) |
| 37 | \( 1 + (-0.556 + 0.964i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.278 - 0.124i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (0.245 + 2.33i)T + (-42.0 + 8.94i)T^{2} \) |
| 47 | \( 1 + (8.64 + 6.28i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (3.24 - 3.60i)T + (-5.54 - 52.7i)T^{2} \) |
| 59 | \( 1 + (1.11 - 0.498i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 + (-0.506 - 0.876i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (8.18 - 9.08i)T + (-7.42 - 70.6i)T^{2} \) |
| 73 | \( 1 + (13.3 - 2.84i)T + (66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (-4.57 - 0.971i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (-0.814 - 0.362i)T + (55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 + (2.40 - 7.39i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-2.36 + 7.27i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.40132632756095377347205990012, −9.833976024268587998304918475831, −8.620766257111760597254972471168, −8.043459879754122324091385131267, −7.25440603755378130868834530151, −6.00551735327283344198303832188, −5.42224457588558332610523015150, −4.73195926268190160968854961590, −2.89005519880305353992488160842, −1.81282155198534725380176600690,
0.34949075409807174822180347737, 1.63181232470431237361131071808, 2.89995896589340042477148144003, 4.50144434943499931031159585096, 5.03915002847039681178487993162, 6.27253734258259928842129344296, 7.47631089113560322853078429420, 7.84051837372243337291074970587, 8.779280999781357219185617820949, 9.842508069132554005136259025804
