| L(s) = 1 | + (−8 − 13.8i)2-s + (137. + 238. i)3-s + (−127. + 221. i)4-s + (−526. − 912. i)5-s + (2.19e3 − 3.80e3i)6-s − 5.02e3·7-s + 4.09e3·8-s + (−2.79e4 + 4.83e4i)9-s + (−8.42e3 + 1.45e4i)10-s − 3.42e4·11-s − 7.03e4·12-s + (7.81e4 − 1.35e5i)13-s + (4.02e4 + 6.96e4i)14-s + (1.44e5 − 2.50e5i)15-s + (−3.27e4 − 5.67e4i)16-s + (−1.40e5 − 2.43e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.979 + 1.69i)3-s + (−0.249 + 0.433i)4-s + (−0.376 − 0.652i)5-s + (0.692 − 1.19i)6-s − 0.791·7-s + 0.353·8-s + (−1.41 + 2.45i)9-s + (−0.266 + 0.461i)10-s − 0.706·11-s − 0.979·12-s + (0.758 − 1.31i)13-s + (0.279 + 0.484i)14-s + (0.738 − 1.27i)15-s + (−0.125 − 0.216i)16-s + (−0.407 − 0.706i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.576i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.816 + 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.0241515 - 0.0760514i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0241515 - 0.0760514i\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 + (3.84e5 - 4.18e5i)T \) |
| good | 3 | \( 1 + (-137. - 238. i)T + (-9.84e3 + 1.70e4i)T^{2} \) |
| 5 | \( 1 + (526. + 912. i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 7 | \( 1 + 5.02e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 3.42e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + (-7.81e4 + 1.35e5i)T + (-5.30e9 - 9.18e9i)T^{2} \) |
| 17 | \( 1 + (1.40e5 + 2.43e5i)T + (-5.92e10 + 1.02e11i)T^{2} \) |
| 23 | \( 1 + (-4.84e5 + 8.38e5i)T + (-9.00e11 - 1.55e12i)T^{2} \) |
| 29 | \( 1 + (3.60e5 - 6.23e5i)T + (-7.25e12 - 1.25e13i)T^{2} \) |
| 31 | \( 1 - 2.63e5T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.85e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + (-4.23e6 - 7.33e6i)T + (-1.63e14 + 2.83e14i)T^{2} \) |
| 43 | \( 1 + (1.92e7 + 3.33e7i)T + (-2.51e14 + 4.35e14i)T^{2} \) |
| 47 | \( 1 + (-6.49e6 + 1.12e7i)T + (-5.59e14 - 9.69e14i)T^{2} \) |
| 53 | \( 1 + (5.06e7 - 8.77e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (-6.26e6 - 1.08e7i)T + (-4.33e15 + 7.50e15i)T^{2} \) |
| 61 | \( 1 + (4.08e7 - 7.07e7i)T + (-5.84e15 - 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-8.99e7 + 1.55e8i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + (-1.37e8 - 2.38e8i)T + (-2.29e16 + 3.97e16i)T^{2} \) |
| 73 | \( 1 + (-6.77e7 - 1.17e8i)T + (-2.94e16 + 5.09e16i)T^{2} \) |
| 79 | \( 1 + (-7.31e7 - 1.26e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + 1.41e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (4.45e8 - 7.72e8i)T + (-1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 + (8.94e7 + 1.54e8i)T + (-3.80e17 + 6.58e17i)T^{2} \) |
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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
−13.79606003470038665381778050073, −12.66515700597261949982850624185, −10.78494075535166434400330532373, −10.12116782595614593814465475683, −8.878362861044939681316373192435, −8.168786784479105239213130833081, −5.12229621513048678931222281902, −3.78801444707966988758913693552, −2.75693204893072912645232029487, −0.02734987256058414348006641210,
1.78311935156759946545187759580, 3.32564509751256661051132030765, 6.41968574536281992405307858431, 6.97904333748492524806700802063, 8.221553892551931848546588950986, 9.233068403852471929281590414467, 11.25184697940519104136689545838, 12.84236456044640473132819461792, 13.58176538170311030582899073988, 14.66527473415030786870363808534
