L(s) = 1 | + 1.41·2-s + 2.00·4-s + (−4.25 − 2.61i)5-s + 4.80i·7-s + 2.82·8-s + (−6.02 − 3.70i)10-s − 4.90i·11-s − 14.9i·13-s + 6.79i·14-s + 4.00·16-s − 6.09·17-s + 14.5·19-s + (−8.51 − 5.23i)20-s − 6.93i·22-s − 42.3·23-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.500·4-s + (−0.851 − 0.523i)5-s + 0.686i·7-s + 0.353·8-s + (−0.602 − 0.370i)10-s − 0.445i·11-s − 1.14i·13-s + 0.485i·14-s + 0.250·16-s − 0.358·17-s + 0.764·19-s + (−0.425 − 0.261i)20-s − 0.315i·22-s − 1.84·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.523 + 0.851i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.523 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.449943747\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.449943747\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (4.25 + 2.61i)T \) |
good | 7 | \( 1 - 4.80iT - 49T^{2} \) |
| 11 | \( 1 + 4.90iT - 121T^{2} \) |
| 13 | \( 1 + 14.9iT - 169T^{2} \) |
| 17 | \( 1 + 6.09T + 289T^{2} \) |
| 19 | \( 1 - 14.5T + 361T^{2} \) |
| 23 | \( 1 + 42.3T + 529T^{2} \) |
| 29 | \( 1 + 40.0iT - 841T^{2} \) |
| 31 | \( 1 + 43.1T + 961T^{2} \) |
| 37 | \( 1 + 49.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 32.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 31.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 4.72T + 2.20e3T^{2} \) |
| 53 | \( 1 + 19.9T + 2.80e3T^{2} \) |
| 59 | \( 1 + 75.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 3.65T + 3.72e3T^{2} \) |
| 67 | \( 1 - 59.3iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 73.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 45.6iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 124.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 83.6T + 6.88e3T^{2} \) |
| 89 | \( 1 - 138. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 10.0iT - 9.40e3T^{2} \) |
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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
−9.797284865851030904934109840775, −8.754886522259069015421792538320, −7.983881254433802936289109576368, −7.28244224523037354512409338294, −5.83520080577434790570457453205, −5.46523755125857926496811990467, −4.19500148425210924607221688425, −3.43078766015306698725161964590, −2.16387612600281780656841257580, −0.36844043241258241789905711188,
1.66861800079543124977722456039, 3.10299842460768236614851222715, 4.05087633582245200085236056494, 4.65881140593071273463521415303, 6.02246156893196835985553829140, 7.00505555832057161853849730740, 7.42260311510045977236162097143, 8.476779980379706656279190614898, 9.677335224951710349004077073066, 10.49863355021099935484897786414