| L(s) = 1 | + (5.80 − 10.0i)2-s + (−13.5 + 7.79i)3-s + (−35.4 − 61.4i)4-s + (−165. − 95.4i)5-s + 181. i·6-s + (−103. − 327. i)7-s − 80.4·8-s + (121.5 − 210. i)9-s + (−1.92e3 + 1.10e3i)10-s + (1.02e3 + 1.77e3i)11-s + (957. + 552. i)12-s − 3.05e3i·13-s + (−3.89e3 − 861. i)14-s + 2.97e3·15-s + (1.80e3 − 3.12e3i)16-s + (2.46e3 − 1.42e3i)17-s + ⋯ |
| L(s) = 1 | + (0.725 − 1.25i)2-s + (−0.5 + 0.288i)3-s + (−0.554 − 0.959i)4-s + (−1.32 − 0.763i)5-s + 0.838i·6-s + (−0.301 − 0.953i)7-s − 0.157·8-s + (0.166 − 0.288i)9-s + (−1.92 + 1.10i)10-s + (0.772 + 1.33i)11-s + (0.554 + 0.319i)12-s − 1.39i·13-s + (−1.41 − 0.313i)14-s + 0.881·15-s + (0.439 − 0.762i)16-s + (0.502 − 0.290i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.178i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.984 + 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.117285 - 1.30676i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.117285 - 1.30676i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (13.5 - 7.79i)T \) |
| 7 | \( 1 + (103. + 327. i)T \) |
| good | 2 | \( 1 + (-5.80 + 10.0i)T + (-32 - 55.4i)T^{2} \) |
| 5 | \( 1 + (165. + 95.4i)T + (7.81e3 + 1.35e4i)T^{2} \) |
| 11 | \( 1 + (-1.02e3 - 1.77e3i)T + (-8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 + 3.05e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + (-2.46e3 + 1.42e3i)T + (1.20e7 - 2.09e7i)T^{2} \) |
| 19 | \( 1 + (3.42e3 + 1.97e3i)T + (2.35e7 + 4.07e7i)T^{2} \) |
| 23 | \( 1 + (330. - 572. i)T + (-7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 + 9.28e3T + 5.94e8T^{2} \) |
| 31 | \( 1 + (-2.42e3 + 1.40e3i)T + (4.43e8 - 7.68e8i)T^{2} \) |
| 37 | \( 1 + (-1.84e4 + 3.19e4i)T + (-1.28e9 - 2.22e9i)T^{2} \) |
| 41 | \( 1 - 6.79e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 1.23e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + (-1.27e5 - 7.38e4i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 + (1.09e5 + 1.90e5i)T + (-1.10e10 + 1.91e10i)T^{2} \) |
| 59 | \( 1 + (-1.66e5 + 9.62e4i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (-2.88e5 - 1.66e5i)T + (2.57e10 + 4.46e10i)T^{2} \) |
| 67 | \( 1 + (1.74e5 + 3.02e5i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 - 3.05e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + (2.04e5 - 1.18e5i)T + (7.56e10 - 1.31e11i)T^{2} \) |
| 79 | \( 1 + (-2.93e5 + 5.07e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + 1.06e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + (-1.13e4 - 6.52e3i)T + (2.48e11 + 4.30e11i)T^{2} \) |
| 97 | \( 1 + 2.05e5iT - 8.32e11T^{2} \) |
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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
−16.20915531880925203816708762557, −14.83632242809148568554049467179, −12.95255916408358650015389048336, −12.26629928900685295506427902189, −11.13376880923519923273809705275, −9.886359299621615824211730514176, −7.54855296551051225826300125306, −4.72998357678611418120450307204, −3.73960761904721515519785837165, −0.74572976032521455474560654063,
3.89926635835319500317444055524, 5.95182799271489753392232297541, 6.97890153111633931684950543511, 8.475523699180177711194601600861, 11.19937206663194487873251450819, 12.21176646723579781760296968359, 13.97743308266528590838191023005, 14.99403567905021707931893845126, 16.04375298198702684754797895112, 16.83057316937563270199023615010
