| 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
−15.00045364790599730123578737726, −14.21909330327833437360341423790, −12.90023347800179037197882092396, −11.21511354609005498061741009489, −9.731005711173974500949743455298, −8.698631490184349890007939994894, −7.58660007033905820509122677366, −5.07077682728135444559259138875, −4.45653395968012521077395412670, −2.78920085881188143698677835832,
0.48723474737076303915771116546, 2.24775316939789714917808115813, 3.12728617614770863168083020453, 5.51817849852979899126012960027, 7.49000038234824674126677371783, 8.010116017338315255347211054221, 10.03765855433335908709235640139, 11.42274526331462880480399659144, 12.71564256366338006829574401461, 13.33636239293376582655636614998
