L(s) = 1 | + (0.984 − 0.173i)2-s + (0.536 + 0.639i)3-s + (0.939 − 0.342i)4-s + (0.639 + 0.536i)6-s + (3.62 + 2.09i)7-s + (0.866 − 0.5i)8-s + (0.399 − 2.26i)9-s + (−2.69 − 4.66i)11-s + (0.723 + 0.417i)12-s + (2.52 − 3.01i)13-s + (3.93 + 1.43i)14-s + (0.766 − 0.642i)16-s + (−2.10 + 0.371i)17-s − 2.30i·18-s + (2.92 + 3.22i)19-s + ⋯ |
L(s) = 1 | + (0.696 − 0.122i)2-s + (0.309 + 0.369i)3-s + (0.469 − 0.171i)4-s + (0.261 + 0.219i)6-s + (1.36 + 0.790i)7-s + (0.306 − 0.176i)8-s + (0.133 − 0.755i)9-s + (−0.812 − 1.40i)11-s + (0.208 + 0.120i)12-s + (0.700 − 0.835i)13-s + (1.05 + 0.382i)14-s + (0.191 − 0.160i)16-s + (−0.511 + 0.0901i)17-s − 0.542i·18-s + (0.671 + 0.740i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.179i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 + 0.179i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.04913 - 0.275985i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.04913 - 0.275985i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.984 + 0.173i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-2.92 - 3.22i)T \) |
good | 3 | \( 1 + (-0.536 - 0.639i)T + (-0.520 + 2.95i)T^{2} \) |
| 7 | \( 1 + (-3.62 - 2.09i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.69 + 4.66i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.52 + 3.01i)T + (-2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (2.10 - 0.371i)T + (15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (0.976 + 2.68i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (1.34 - 7.65i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (2.86 - 4.96i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 1.84iT - 37T^{2} \) |
| 41 | \( 1 + (-4.67 + 3.92i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (4.31 - 11.8i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-4.31 - 0.760i)T + (44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (4.44 + 12.2i)T + (-40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (-1.36 - 7.75i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (10.3 - 3.74i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-10.3 - 1.82i)T + (62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (12.6 + 4.61i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (2.16 + 2.58i)T + (-12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-3.96 + 3.32i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (7.56 + 4.36i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.52 - 2.12i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (0.851 - 0.150i)T + (91.1 - 33.1i)T^{2} \) |
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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
−10.29004273826819175209931211948, −8.925563669448393867259552905336, −8.462691117948170585260443670174, −7.65167888825730539923422739196, −6.24107620213546491000677946812, −5.55367039447521191206655837652, −4.80674481865165058496533402175, −3.55939357490001390395604708388, −2.86400128140914803963496585154, −1.35187573138042359542908175886,
1.67268127349796911662428544100, 2.38691433065067990649133038369, 4.12135462060725161367373378607, 4.63814196733497153447664460696, 5.50535398853281782836182450941, 6.89394724464847982749356215041, 7.58226409744840307717222881068, 7.928554934964088901385881259733, 9.169344780967537158854669711740, 10.25860423807400498167366993962