L(s) = 1 | + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (1.75 + 3.03i)5-s + (2.30 − 1.29i)7-s − 0.999i·8-s + (3.03 + 1.75i)10-s + (4.72 + 2.72i)11-s − 2.51i·13-s + (1.35 − 2.27i)14-s + (−0.5 − 0.866i)16-s + (−0.852 + 1.47i)17-s + (−4.19 + 2.42i)19-s + 3.50·20-s + 5.45·22-s + (−4.63 + 2.67i)23-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (0.784 + 1.35i)5-s + (0.872 − 0.487i)7-s − 0.353i·8-s + (0.960 + 0.554i)10-s + (1.42 + 0.822i)11-s − 0.698i·13-s + (0.362 − 0.607i)14-s + (−0.125 − 0.216i)16-s + (−0.206 + 0.358i)17-s + (−0.963 + 0.556i)19-s + 0.784·20-s + 1.16·22-s + (−0.966 + 0.558i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.112i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.021595771\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.021595771\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.30 + 1.29i)T \) |
good | 5 | \( 1 + (-1.75 - 3.03i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-4.72 - 2.72i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 2.51iT - 13T^{2} \) |
| 17 | \( 1 + (0.852 - 1.47i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4.19 - 2.42i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.63 - 2.67i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 7.12iT - 29T^{2} \) |
| 31 | \( 1 + (5.25 + 3.03i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.14 - 7.18i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 4.75T + 41T^{2} \) |
| 43 | \( 1 - 6.45T + 43T^{2} \) |
| 47 | \( 1 + (4.31 + 7.46i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-10.0 - 5.81i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.94 + 6.83i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (7.26 - 4.19i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.895 + 1.55i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 0.791iT - 71T^{2} \) |
| 73 | \( 1 + (11.5 + 6.66i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.40 + 4.17i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 0.465T + 83T^{2} \) |
| 89 | \( 1 + (-1.09 - 1.89i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 4.89iT - 97T^{2} \) |
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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
−10.14174698950144575426028084185, −9.319643619188097247345846175638, −7.988529286626936517385046111358, −7.18099411202412157056722229589, −6.30469141656934179246597470513, −5.77133859874161060944622339545, −4.35013413551917151659158026827, −3.77186677220889835043558132773, −2.38939879587358630529344962115, −1.63683661410655732533154616470,
1.29978140983617055589798334494, 2.30894115404906004570170759853, 4.03557449317919696822022223565, 4.61796404976065960886190041730, 5.58198684276465735778608221928, 6.15323145609560227072836295728, 7.16757407821414584619906232198, 8.463594602799650775499211663902, 8.872319197795821964236277915852, 9.372977444441237557215275496801