| 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 | \( 1 + (-10.2 + 7.42i)T + (29.9 - 92.2i)T^{2} \) |
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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
−9.921701754896641586349102880186, −9.128724209280242971995740509769, −8.001523027062651684941714365126, −7.14753223823253795550805584124, −6.31479491946493985386965001320, −5.11316874289037024044042384629, −4.21895028826830070124096777469, −3.20377353099288645293411605362, −1.94178903419024536809384449000, −0.19227561527600727662373566229,
1.42027112666139585942505497094, 3.30172193315504539484497654809, 4.48467439102017092122469231541, 5.37764258657713500761655274628, 6.28047179431048278030910642626, 6.83532780619185787434723304692, 8.128481817022217586705201008268, 8.485353136940905981938257777579, 9.594350015431149297128777302380, 10.32726719842278699831379783429
