| L(s) = 1 | + (8 − 13.8i)2-s + (116. − 202. i)3-s + (−127. − 221. i)4-s + (−509. + 882. i)5-s + (−1.86e3 − 3.23e3i)6-s + 2.72e3·7-s − 4.09e3·8-s + (−1.74e4 − 3.01e4i)9-s + (8.15e3 + 1.41e4i)10-s − 8.35e4·11-s − 5.97e4·12-s + (−8.43e4 − 1.46e5i)13-s + (2.18e4 − 3.77e4i)14-s + (1.19e5 + 2.06e5i)15-s + (−3.27e4 + 5.67e4i)16-s + (9.35e3 − 1.62e4i)17-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (0.832 − 1.44i)3-s + (−0.249 − 0.433i)4-s + (−0.364 + 0.631i)5-s + (−0.588 − 1.01i)6-s + 0.429·7-s − 0.353·8-s + (−0.885 − 1.53i)9-s + (0.257 + 0.446i)10-s − 1.72·11-s − 0.832·12-s + (−0.818 − 1.41i)13-s + (0.151 − 0.262i)14-s + (0.607 + 1.05i)15-s + (−0.125 + 0.216i)16-s + (0.0271 − 0.0470i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.897 - 0.441i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.897 - 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.397706 + 1.71011i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.397706 + 1.71011i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-8 + 13.8i)T \) |
| 19 | \( 1 + (-4.45e5 - 3.52e5i)T \) |
| good | 3 | \( 1 + (-116. + 202. i)T + (-9.84e3 - 1.70e4i)T^{2} \) |
| 5 | \( 1 + (509. - 882. i)T + (-9.76e5 - 1.69e6i)T^{2} \) |
| 7 | \( 1 - 2.72e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 8.35e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + (8.43e4 + 1.46e5i)T + (-5.30e9 + 9.18e9i)T^{2} \) |
| 17 | \( 1 + (-9.35e3 + 1.62e4i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 23 | \( 1 + (-1.45e5 - 2.52e5i)T + (-9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 + (3.58e6 + 6.20e6i)T + (-7.25e12 + 1.25e13i)T^{2} \) |
| 31 | \( 1 - 9.45e5T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.52e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + (-8.09e6 + 1.40e7i)T + (-1.63e14 - 2.83e14i)T^{2} \) |
| 43 | \( 1 + (-3.78e6 + 6.55e6i)T + (-2.51e14 - 4.35e14i)T^{2} \) |
| 47 | \( 1 + (1.77e7 + 3.07e7i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 + (-5.01e6 - 8.68e6i)T + (-1.64e15 + 2.85e15i)T^{2} \) |
| 59 | \( 1 + (-8.10e7 + 1.40e8i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (-3.58e7 - 6.20e7i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (2.98e7 + 5.16e7i)T + (-1.36e16 + 2.35e16i)T^{2} \) |
| 71 | \( 1 + (5.33e7 - 9.23e7i)T + (-2.29e16 - 3.97e16i)T^{2} \) |
| 73 | \( 1 + (3.84e7 - 6.65e7i)T + (-2.94e16 - 5.09e16i)T^{2} \) |
| 79 | \( 1 + (-1.25e8 + 2.18e8i)T + (-5.99e16 - 1.03e17i)T^{2} \) |
| 83 | \( 1 + 5.74e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (7.10e7 + 1.23e8i)T + (-1.75e17 + 3.03e17i)T^{2} \) |
| 97 | \( 1 + (-5.96e7 + 1.03e8i)T + (-3.80e17 - 6.58e17i)T^{2} \) |
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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
−13.33636239293376582655636614998, −12.71564256366338006829574401461, −11.42274526331462880480399659144, −10.03765855433335908709235640139, −8.010116017338315255347211054221, −7.49000038234824674126677371783, −5.51817849852979899126012960027, −3.12728617614770863168083020453, −2.24775316939789714917808115813, −0.48723474737076303915771116546,
2.78920085881188143698677835832, 4.45653395968012521077395412670, 5.07077682728135444559259138875, 7.58660007033905820509122677366, 8.698631490184349890007939994894, 9.731005711173974500949743455298, 11.21511354609005498061741009489, 12.90023347800179037197882092396, 14.21909330327833437360341423790, 15.00045364790599730123578737726
