| L(s) = 1 | + (8 + 13.8i)2-s + (6.53 + 11.3i)3-s + (−127. + 221. i)4-s + (80.6 + 139. i)5-s + (−104. + 181. i)6-s − 6.27e3·7-s − 4.09e3·8-s + (9.75e3 − 1.68e4i)9-s + (−1.28e3 + 2.23e3i)10-s − 2.19e4·11-s − 3.34e3·12-s + (−1.18e4 + 2.04e4i)13-s + (−5.01e4 − 8.68e4i)14-s + (−1.05e3 + 1.82e3i)15-s + (−3.27e4 − 5.67e4i)16-s + (−1.82e5 − 3.16e5i)17-s + ⋯ |
| L(s) = 1 | + (0.353 + 0.612i)2-s + (0.0465 + 0.0806i)3-s + (−0.249 + 0.433i)4-s + (0.0576 + 0.0999i)5-s + (−0.0329 + 0.0570i)6-s − 0.987·7-s − 0.353·8-s + (0.495 − 0.858i)9-s + (−0.0407 + 0.0706i)10-s − 0.452·11-s − 0.0465·12-s + (−0.114 + 0.198i)13-s + (−0.348 − 0.604i)14-s + (−0.00537 + 0.00930i)15-s + (−0.125 − 0.216i)16-s + (−0.530 − 0.918i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0235 + 0.999i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.0235 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.554360 - 0.541442i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.554360 - 0.541442i\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 + (1.33e5 + 5.52e5i)T \) |
| good | 3 | \( 1 + (-6.53 - 11.3i)T + (-9.84e3 + 1.70e4i)T^{2} \) |
| 5 | \( 1 + (-80.6 - 139. i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 7 | \( 1 + 6.27e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 2.19e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + (1.18e4 - 2.04e4i)T + (-5.30e9 - 9.18e9i)T^{2} \) |
| 17 | \( 1 + (1.82e5 + 3.16e5i)T + (-5.92e10 + 1.02e11i)T^{2} \) |
| 23 | \( 1 + (5.74e5 - 9.94e5i)T + (-9.00e11 - 1.55e12i)T^{2} \) |
| 29 | \( 1 + (-3.46e6 + 6.00e6i)T + (-7.25e12 - 1.25e13i)T^{2} \) |
| 31 | \( 1 + 1.02e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.26e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + (3.30e6 + 5.72e6i)T + (-1.63e14 + 2.83e14i)T^{2} \) |
| 43 | \( 1 + (6.69e6 + 1.16e7i)T + (-2.51e14 + 4.35e14i)T^{2} \) |
| 47 | \( 1 + (1.72e7 - 2.98e7i)T + (-5.59e14 - 9.69e14i)T^{2} \) |
| 53 | \( 1 + (4.48e7 - 7.77e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (-3.01e7 - 5.22e7i)T + (-4.33e15 + 7.50e15i)T^{2} \) |
| 61 | \( 1 + (8.01e7 - 1.38e8i)T + (-5.84e15 - 1.01e16i)T^{2} \) |
| 67 | \( 1 + (5.27e6 - 9.14e6i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + (-4.02e7 - 6.97e7i)T + (-2.29e16 + 3.97e16i)T^{2} \) |
| 73 | \( 1 + (-6.50e7 - 1.12e8i)T + (-2.94e16 + 5.09e16i)T^{2} \) |
| 79 | \( 1 + (2.84e8 + 4.91e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + 5.01e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (-2.35e8 + 4.07e8i)T + (-1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 + (1.71e8 + 2.97e8i)T + (-3.80e17 + 6.58e17i)T^{2} \) |
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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
−13.97229094006788041409389666683, −13.00567696554863687999519022096, −11.86416515179245185530352055437, −10.04178765002472779151084959133, −8.956218895279916820751488975270, −7.19310684980265454891073469587, −6.20998990215268891168161512366, −4.45908113702251822198702705807, −2.90764634952274908534103279740, −0.23877665349747503045790958693,
1.78218195325426928155738320250, 3.38316575584333023210925511638, 5.01439956913827396074670333820, 6.61359365756091417491648312971, 8.375540774590790112356013083801, 9.981380222515578751624313006872, 10.78214562873185386556181088667, 12.56241523234854736758768247071, 13.04895525763101562523058933496, 14.36128878005154133628017518372
