| L(s) = 1 | + (16.3 + 11.8i)3-s + (18.3 − 56.4i)5-s + (89.0 − 64.6i)7-s + (51.4 + 158. i)9-s + (161. − 367. i)11-s + (−359. − 1.10e3i)13-s + (971. − 705. i)15-s + (−621. + 1.91e3i)17-s + (1.72e3 + 1.25e3i)19-s + 2.22e3·21-s − 1.27e3·23-s + (−321. − 233. i)25-s + (479. − 1.47e3i)27-s + (4.86e3 − 3.53e3i)29-s + (1.22e3 + 3.75e3i)31-s + ⋯ |
| L(s) = 1 | + (1.05 + 0.762i)3-s + (0.328 − 1.00i)5-s + (0.686 − 0.499i)7-s + (0.211 + 0.651i)9-s + (0.401 − 0.915i)11-s + (−0.589 − 1.81i)13-s + (1.11 − 0.809i)15-s + (−0.521 + 1.60i)17-s + (1.09 + 0.797i)19-s + 1.10·21-s − 0.501·23-s + (−0.102 − 0.0747i)25-s + (0.126 − 0.389i)27-s + (1.07 − 0.780i)29-s + (0.228 + 0.702i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.896 + 0.442i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.896 + 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.74891 - 0.641714i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.74891 - 0.641714i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + (-161. + 367. i)T \) |
| good | 3 | \( 1 + (-16.3 - 11.8i)T + (75.0 + 231. i)T^{2} \) |
| 5 | \( 1 + (-18.3 + 56.4i)T + (-2.52e3 - 1.83e3i)T^{2} \) |
| 7 | \( 1 + (-89.0 + 64.6i)T + (5.19e3 - 1.59e4i)T^{2} \) |
| 13 | \( 1 + (359. + 1.10e3i)T + (-3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (621. - 1.91e3i)T + (-1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (-1.72e3 - 1.25e3i)T + (7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 + 1.27e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + (-4.86e3 + 3.53e3i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-1.22e3 - 3.75e3i)T + (-2.31e7 + 1.68e7i)T^{2} \) |
| 37 | \( 1 + (5.18e3 - 3.76e3i)T + (2.14e7 - 6.59e7i)T^{2} \) |
| 41 | \( 1 + (-1.32e4 - 9.62e3i)T + (3.58e7 + 1.10e8i)T^{2} \) |
| 43 | \( 1 + 8.87e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (6.36e3 + 4.62e3i)T + (7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-4.27e3 - 1.31e4i)T + (-3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-6.94e3 + 5.04e3i)T + (2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (1.84e3 - 5.66e3i)T + (-6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 + 5.26e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (3.85e3 - 1.18e4i)T + (-1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-1.05e3 + 767. i)T + (6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-5.73e3 - 1.76e4i)T + (-2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (5.32e3 - 1.63e4i)T + (-3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 + 1.27e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + (7.60e3 + 2.34e4i)T + (-6.94e9 + 5.04e9i)T^{2} \) |
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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
−13.33419046917513274719790628673, −12.20388075025991802172865520428, −10.60332589043381371800139941456, −9.772153331016266538995066186214, −8.449880740260217522294067776003, −8.076130842102522265343072514570, −5.76770947434614881556246307117, −4.45736013417285662570178971804, −3.17999506556829984251411141291, −1.16155344117666988449224204109,
1.90823480891618057284360074116, 2.71140700484184456484016828175, 4.74869656946415735998352496125, 6.87440809923420031251515565838, 7.28721591687716750513324501649, 8.866915503038917115693277962492, 9.661805046462643701127746752310, 11.35648967503100451302566366550, 12.13526633677573257828707419935, 13.76831587359688842134434847236
