| L(s) = 1 | − 5.65·2-s + 15.5·3-s + 32.0·4-s − 29.6i·5-s − 88.1·6-s + 53.4i·7-s − 181.·8-s + 243·9-s + 167. i·10-s + 274. i·11-s + 498.·12-s − 650.·13-s − 302. i·14-s − 461. i·15-s + 1.02e3·16-s + 4.70e3i·17-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.500·4-s − 0.237i·5-s − 0.408·6-s + 0.155i·7-s − 0.353·8-s + 0.333·9-s + 0.167i·10-s + 0.206i·11-s + 0.288·12-s − 0.295·13-s − 0.110i·14-s − 0.136i·15-s + 0.250·16-s + 0.957i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.381 - 0.924i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.381 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.468282842\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.468282842\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 5.65T \) |
| 3 | \( 1 - 15.5T \) |
| 23 | \( 1 + (-4.63e3 + 1.12e4i)T \) |
| good | 5 | \( 1 + 29.6iT - 1.56e4T^{2} \) |
| 7 | \( 1 - 53.4iT - 1.17e5T^{2} \) |
| 11 | \( 1 - 274. iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 650.T + 4.82e6T^{2} \) |
| 17 | \( 1 - 4.70e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 7.07e3iT - 4.70e7T^{2} \) |
| 29 | \( 1 + 3.37e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + 3.88e3T + 8.87e8T^{2} \) |
| 37 | \( 1 + 6.69e3iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 6.32e4T + 4.75e9T^{2} \) |
| 43 | \( 1 - 1.44e5iT - 6.32e9T^{2} \) |
| 47 | \( 1 + 1.92e4T + 1.07e10T^{2} \) |
| 53 | \( 1 - 2.00e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 2.82e5T + 4.21e10T^{2} \) |
| 61 | \( 1 - 3.82e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 4.47e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 2.20e5T + 1.28e11T^{2} \) |
| 73 | \( 1 - 2.75e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 5.01e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 2.40e4iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 3.05e4iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 3.67e5iT - 8.32e11T^{2} \) |
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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
−12.33700132883734538095863558649, −10.98824623785314384584570235643, −10.03530025377004141327823593784, −9.029601988795631957909793793128, −8.191468350192089344630176639801, −7.15701416978739959145347739282, −5.80562663387110248602861487376, −4.13592780860079087770197763933, −2.59808793411471191411229410429, −1.26389980609878902935369963021,
0.56730525170548901719921180140, 2.21463031263002130397367401093, 3.44614355957004665428066230900, 5.18473311637196826764534009309, 6.86379444210229583146358515390, 7.58018834188330766058028086053, 8.884042098856603766004790806759, 9.554515172924528850425207493447, 10.75367491071123943585858207066, 11.61043143034264015630553067679
