L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (1.26 − 1.84i)5-s + (2.17 + 3.76i)7-s + 0.999·8-s + (0.960 + 2.01i)10-s + (2.04 + 1.17i)11-s + (3.18 + 1.69i)13-s − 4.34·14-s + (−0.5 + 0.866i)16-s + (2.60 − 1.50i)17-s + (0.585 − 0.338i)19-s + (−2.22 − 0.178i)20-s + (−2.04 + 1.17i)22-s + (−5.58 − 3.22i)23-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.567 − 0.823i)5-s + (0.821 + 1.42i)7-s + 0.353·8-s + (0.303 + 0.638i)10-s + (0.615 + 0.355i)11-s + (0.883 + 0.469i)13-s − 1.16·14-s + (−0.125 + 0.216i)16-s + (0.631 − 0.364i)17-s + (0.134 − 0.0776i)19-s + (−0.498 − 0.0398i)20-s + (−0.435 + 0.251i)22-s + (−1.16 − 0.672i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.527 - 0.849i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.527 - 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.730521858\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.730521858\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.26 + 1.84i)T \) |
| 13 | \( 1 + (-3.18 - 1.69i)T \) |
good | 7 | \( 1 + (-2.17 - 3.76i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.04 - 1.17i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-2.60 + 1.50i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.585 + 0.338i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (5.58 + 3.22i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (4.82 - 8.35i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 7.11iT - 31T^{2} \) |
| 37 | \( 1 + (3.74 - 6.48i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.60 - 1.50i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.91 + 3.41i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 5.61T + 47T^{2} \) |
| 53 | \( 1 - 9.43iT - 53T^{2} \) |
| 59 | \( 1 + (-4.56 + 2.63i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.15 - 3.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.91 + 5.04i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (2.52 - 1.45i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 7.67T + 73T^{2} \) |
| 79 | \( 1 + 3.74T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 + (4.15 + 2.39i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-8.17 - 14.1i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.483078975793674831749765000489, −9.027997540385703604602376438042, −8.448717715823385475156887882969, −7.62693493884168993952467719316, −6.38105459477887741725476017439, −5.72576182921515193229493250569, −5.05987712554295647117515461752, −4.05572256544064637328021343571, −2.25511746882705835962419891592, −1.33386036518228345826633129754,
1.02545373917950853532958149112, 2.04354776462056604317018345182, 3.60575716641379022549857231296, 3.92245488359567365323947814210, 5.45161950709502645031405942365, 6.33605654446002867492062935116, 7.43290110821373570757020910488, 7.900428157166734575129635006699, 8.942339767660978196851115421256, 9.971070837165985051244563078127