L(s) = 1 | + (−1.13 − 0.847i)2-s + (−1.71 + 0.242i)3-s + (0.562 + 1.91i)4-s + (2.14 + 1.17i)6-s − 3.08·7-s + (0.990 − 2.64i)8-s + (2.88 − 0.831i)9-s − 2.54i·11-s + (−1.42 − 3.15i)12-s + 5.06·13-s + (3.49 + 2.61i)14-s + (−3.36 + 2.15i)16-s − 0.214·17-s + (−3.96 − 1.50i)18-s − 2.60·19-s + ⋯ |
L(s) = 1 | + (−0.800 − 0.599i)2-s + (−0.990 + 0.139i)3-s + (0.281 + 0.959i)4-s + (0.876 + 0.481i)6-s − 1.16·7-s + (0.350 − 0.936i)8-s + (0.960 − 0.277i)9-s − 0.767i·11-s + (−0.412 − 0.910i)12-s + 1.40·13-s + (0.934 + 0.700i)14-s + (−0.841 + 0.539i)16-s − 0.0519·17-s + (−0.935 − 0.354i)18-s − 0.598·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.243 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.243 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.141939 + 0.182037i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.141939 + 0.182037i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.13 + 0.847i)T \) |
| 3 | \( 1 + (1.71 - 0.242i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 3.08T + 7T^{2} \) |
| 11 | \( 1 + 2.54iT - 11T^{2} \) |
| 13 | \( 1 - 5.06T + 13T^{2} \) |
| 17 | \( 1 + 0.214T + 17T^{2} \) |
| 19 | \( 1 + 2.60T + 19T^{2} \) |
| 23 | \( 1 - 4.47iT - 23T^{2} \) |
| 29 | \( 1 + 7.86T + 29T^{2} \) |
| 31 | \( 1 - 4.58iT - 31T^{2} \) |
| 37 | \( 1 + 7.67T + 37T^{2} \) |
| 41 | \( 1 - 9.26iT - 41T^{2} \) |
| 43 | \( 1 + 11.4iT - 43T^{2} \) |
| 47 | \( 1 - 10.5iT - 47T^{2} \) |
| 53 | \( 1 - 9.51iT - 53T^{2} \) |
| 59 | \( 1 + 0.428iT - 59T^{2} \) |
| 61 | \( 1 - 1.11iT - 61T^{2} \) |
| 67 | \( 1 - 2.35iT - 67T^{2} \) |
| 71 | \( 1 + 6.12T + 71T^{2} \) |
| 73 | \( 1 - 12.0iT - 73T^{2} \) |
| 79 | \( 1 - 11.6iT - 79T^{2} \) |
| 83 | \( 1 - 2.29T + 83T^{2} \) |
| 89 | \( 1 + 12.4iT - 89T^{2} \) |
| 97 | \( 1 + 8.04iT - 97T^{2} \) |
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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
−10.93421754416470907540975704263, −10.18069954544464290584345343226, −9.310769324421700632670691794309, −8.580218349088698006472501861952, −7.30876435382513018963923519174, −6.43850936027529798588053925582, −5.69053895379659696992389454417, −3.99517588209696106170992288586, −3.24848437875122183110713522740, −1.35106823979007025552851233600,
0.20243571079126867607005771108, 1.88469877273354047624407649563, 3.89058202514979726055478454132, 5.21406952125556143227348705041, 6.22457425305496132465533985614, 6.63462753838885422792904148854, 7.57936150920474015971848821317, 8.734224339939524501908025670695, 9.587484276130257400607746438633, 10.38396872608762146076011417158