L(s) = 1 | + 1.90i·2-s + (5.08 − 1.07i)3-s + 4.35·4-s − 1.24·5-s + (2.05 + 9.70i)6-s + 23.5i·8-s + (24.6 − 10.9i)9-s − 2.38i·10-s + 40.6i·11-s + (22.1 − 4.69i)12-s − 19.5i·13-s + (−6.34 + 1.34i)15-s − 10.1·16-s + 104.·17-s + (20.9 + 47.0i)18-s − 40.4i·19-s + ⋯ |
L(s) = 1 | + 0.674i·2-s + (0.978 − 0.207i)3-s + 0.544·4-s − 0.111·5-s + (0.140 + 0.660i)6-s + 1.04i·8-s + (0.913 − 0.406i)9-s − 0.0752i·10-s + 1.11i·11-s + (0.532 − 0.113i)12-s − 0.418i·13-s + (−0.109 + 0.0231i)15-s − 0.158·16-s + 1.49·17-s + (0.274 + 0.616i)18-s − 0.488i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 - 0.797i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.603 - 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.41812 + 1.20229i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.41812 + 1.20229i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-5.08 + 1.07i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 1.90iT - 8T^{2} \) |
| 5 | \( 1 + 1.24T + 125T^{2} \) |
| 11 | \( 1 - 40.6iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 19.5iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 104.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 40.4iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 80.4iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 211. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 100. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 189.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 186.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 158.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 358.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 423. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 625.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 807. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 298.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 455. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 501. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 61.9T + 4.93e5T^{2} \) |
| 83 | \( 1 + 73.1T + 5.71e5T^{2} \) |
| 89 | \( 1 + 114.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.41e3iT - 9.12e5T^{2} \) |
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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
−12.74851475524238836445542292774, −11.98987923491289114058636415921, −10.52475200297306339456175832803, −9.515831868297571701338804487112, −8.190851573757592475578351149871, −7.51234853663699067284623015225, −6.57666140000918656083693043207, −5.04630227575658032713222513779, −3.29578082019675778562190280430, −1.86676518890572452856189722313,
1.48429823417206950231400203214, 3.00362468584593911877343926230, 3.88002577213761281613107193604, 5.86105061085640865456517924124, 7.34987610288393528786273077513, 8.272766600901585193667570808363, 9.582031406829580329498950410426, 10.31943302629787029909550132683, 11.46856450496218195207552932519, 12.29614800007504265900496382341