L(s) = 1 | + (−0.576 + 1.29i)2-s + 2.50i·3-s + (−1.33 − 1.48i)4-s + (−0.639 − 2.14i)5-s + (−3.23 − 1.44i)6-s + (2.69 − 0.867i)8-s − 3.25·9-s + (3.13 + 0.408i)10-s + 2.25i·11-s + (3.72 − 3.34i)12-s + 5.96·13-s + (5.35 − 1.60i)15-s + (−0.430 + 3.97i)16-s − 2.00·17-s + (1.87 − 4.20i)18-s + 7.81·19-s + ⋯ |
L(s) = 1 | + (−0.407 + 0.913i)2-s + 1.44i·3-s + (−0.667 − 0.744i)4-s + (−0.286 − 0.958i)5-s + (−1.31 − 0.588i)6-s + (0.951 − 0.306i)8-s − 1.08·9-s + (0.991 + 0.129i)10-s + 0.678i·11-s + (1.07 − 0.964i)12-s + 1.65·13-s + (1.38 − 0.413i)15-s + (−0.107 + 0.994i)16-s − 0.486·17-s + (0.442 − 0.991i)18-s + 1.79·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.760 - 0.649i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.760 - 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.407452 + 1.10531i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.407452 + 1.10531i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.576 - 1.29i)T \) |
| 5 | \( 1 + (0.639 + 2.14i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 2.50iT - 3T^{2} \) |
| 11 | \( 1 - 2.25iT - 11T^{2} \) |
| 13 | \( 1 - 5.96T + 13T^{2} \) |
| 17 | \( 1 + 2.00T + 17T^{2} \) |
| 19 | \( 1 - 7.81T + 19T^{2} \) |
| 23 | \( 1 - 2.99T + 23T^{2} \) |
| 29 | \( 1 + 4.87T + 29T^{2} \) |
| 31 | \( 1 + 1.49T + 31T^{2} \) |
| 37 | \( 1 - 4.78iT - 37T^{2} \) |
| 41 | \( 1 + 8.82iT - 41T^{2} \) |
| 43 | \( 1 + 1.12T + 43T^{2} \) |
| 47 | \( 1 - 9.56iT - 47T^{2} \) |
| 53 | \( 1 - 7.06iT - 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 - 1.21iT - 61T^{2} \) |
| 67 | \( 1 + 1.11T + 67T^{2} \) |
| 71 | \( 1 - 8.40iT - 71T^{2} \) |
| 73 | \( 1 - 5.88T + 73T^{2} \) |
| 79 | \( 1 - 12.1iT - 79T^{2} \) |
| 83 | \( 1 + 11.1iT - 83T^{2} \) |
| 89 | \( 1 + 4.57iT - 89T^{2} \) |
| 97 | \( 1 - 4.62T + 97T^{2} \) |
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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
−10.01705193145367102273870812492, −9.234050672562334405906384810720, −8.943151346800345382661596035444, −7.995325493230406854341555087259, −7.05678834203806480341090517039, −5.73917467074870255314089739087, −5.17350343678757658097636874278, −4.30685091991828319438610946256, −3.59672902805154368987738040772, −1.22324403426575395322150615124,
0.77854915055627296945803796158, 1.89092556129249253904233998152, 3.08772672103448295626279392186, 3.73988144556045483989581192208, 5.50266998576347279961519653183, 6.53763232998303897338334883091, 7.28992128339399786837357626187, 8.015825821533483768281878556821, 8.721017315198415412682623175790, 9.726157469978032077222413927251