L(s) = 1 | + (0.823 + 1.42i)3-s − i·5-s + (−1.33 − 0.769i)7-s + (0.143 − 0.249i)9-s + (−2.55 + 1.47i)11-s + (1.44 + 3.30i)13-s + (1.42 − 0.823i)15-s + (2.39 − 4.15i)17-s + (5.26 + 3.03i)19-s − 2.53i·21-s + (4.15 + 7.19i)23-s − 25-s + 5.41·27-s + (4.30 + 7.45i)29-s + 6.96i·31-s + ⋯ |
L(s) = 1 | + (0.475 + 0.823i)3-s − 0.447i·5-s + (−0.503 − 0.290i)7-s + (0.0479 − 0.0830i)9-s + (−0.769 + 0.444i)11-s + (0.400 + 0.916i)13-s + (0.368 − 0.212i)15-s + (0.581 − 1.00i)17-s + (1.20 + 0.696i)19-s − 0.552i·21-s + (0.865 + 1.49i)23-s − 0.200·25-s + 1.04·27-s + (0.798 + 1.38i)29-s + 1.25i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 - 0.797i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.603 - 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.835767747\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.835767747\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 13 | \( 1 + (-1.44 - 3.30i)T \) |
good | 3 | \( 1 + (-0.823 - 1.42i)T + (-1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + (1.33 + 0.769i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.55 - 1.47i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-2.39 + 4.15i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.26 - 3.03i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.15 - 7.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.30 - 7.45i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 6.96iT - 31T^{2} \) |
| 37 | \( 1 + (3.70 - 2.13i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-10.2 + 5.89i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.53 + 4.39i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 6.64iT - 47T^{2} \) |
| 53 | \( 1 + 5.71T + 53T^{2} \) |
| 59 | \( 1 + (-1.08 - 0.625i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.24 + 9.09i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (9.25 - 5.34i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (8.09 + 4.67i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 10.1iT - 73T^{2} \) |
| 79 | \( 1 + 7.47T + 79T^{2} \) |
| 83 | \( 1 + 6.44iT - 83T^{2} \) |
| 89 | \( 1 + (-2.70 + 1.56i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (3.91 + 2.25i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.917500754424143042643146326212, −9.273697666587740956186113902323, −8.713270930170475743098890479352, −7.45561916606623568577055007099, −6.91122089995255767863230824451, −5.44972075948912769056866700933, −4.84802042484823642370386925152, −3.66981336416594453845599616401, −3.07202396979428010746838710682, −1.30687765496368537700302505010,
0.928143651124725747107566908124, 2.63138767545204172877636432901, 2.99507574381760971789940712995, 4.51671405802203863549195870918, 5.81124840983634955946547196718, 6.36025547309867559316366203141, 7.59698438925700330427331480189, 7.891497665064487096056529630939, 8.810072022344531085278630851011, 9.862309516406413232024259346861