L(s) = 1 | + (−0.309 + 0.951i)2-s + (−0.978 − 0.207i)3-s + (−0.809 − 0.587i)4-s + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)6-s + (−0.897 + 0.399i)7-s + (0.809 − 0.587i)8-s + (0.913 + 0.406i)9-s + (0.978 − 0.207i)10-s + (0.250 − 2.38i)11-s + (0.669 + 0.743i)12-s + (−0.151 + 0.168i)13-s + (−0.102 − 0.976i)14-s + (0.309 + 0.951i)15-s + (0.309 + 0.951i)16-s + (0.443 + 4.21i)17-s + ⋯ |
L(s) = 1 | + (−0.218 + 0.672i)2-s + (−0.564 − 0.120i)3-s + (−0.404 − 0.293i)4-s + (−0.223 − 0.387i)5-s + (0.204 − 0.353i)6-s + (−0.339 + 0.151i)7-s + (0.286 − 0.207i)8-s + (0.304 + 0.135i)9-s + (0.309 − 0.0657i)10-s + (0.0754 − 0.717i)11-s + (0.193 + 0.214i)12-s + (−0.0420 + 0.0467i)13-s + (−0.0274 − 0.261i)14-s + (0.0797 + 0.245i)15-s + (0.0772 + 0.237i)16-s + (0.107 + 1.02i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.961 - 0.275i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.961 - 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0473060 + 0.336347i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0473060 + 0.336347i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.309 - 0.951i)T \) |
| 3 | \( 1 + (0.978 + 0.207i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 + (-5.35 - 1.52i)T \) |
good | 7 | \( 1 + (0.897 - 0.399i)T + (4.68 - 5.20i)T^{2} \) |
| 11 | \( 1 + (-0.250 + 2.38i)T + (-10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (0.151 - 0.168i)T + (-1.35 - 12.9i)T^{2} \) |
| 17 | \( 1 + (-0.443 - 4.21i)T + (-16.6 + 3.53i)T^{2} \) |
| 19 | \( 1 + (1.63 + 1.81i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (2.11 - 1.54i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (0.494 - 1.52i)T + (-23.4 - 17.0i)T^{2} \) |
| 37 | \( 1 + (4.86 - 8.42i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (11.6 - 2.46i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (-1.56 - 1.73i)T + (-4.49 + 42.7i)T^{2} \) |
| 47 | \( 1 + (0.265 + 0.818i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (10.0 + 4.46i)T + (35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (6.52 + 1.38i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 - 3.26T + 61T^{2} \) |
| 67 | \( 1 + (-4.50 - 7.80i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (3.43 + 1.52i)T + (47.5 + 52.7i)T^{2} \) |
| 73 | \( 1 + (0.974 - 9.27i)T + (-71.4 - 15.1i)T^{2} \) |
| 79 | \( 1 + (-0.770 - 7.33i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (-1.65 + 0.352i)T + (75.8 - 33.7i)T^{2} \) |
| 89 | \( 1 + (10.0 + 7.28i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-10.2 - 7.42i)T + (29.9 + 92.2i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.32726719842278699831379783429, −9.594350015431149297128777302380, −8.485353136940905981938257777579, −8.128481817022217586705201008268, −6.83532780619185787434723304692, −6.28047179431048278030910642626, −5.37764258657713500761655274628, −4.48467439102017092122469231541, −3.30172193315504539484497654809, −1.42027112666139585942505497094,
0.19227561527600727662373566229, 1.94178903419024536809384449000, 3.20377353099288645293411605362, 4.21895028826830070124096777469, 5.11316874289037024044042384629, 6.31479491946493985386965001320, 7.14753223823253795550805584124, 8.001523027062651684941714365126, 9.128724209280242971995740509769, 9.921701754896641586349102880186