| L(s) = 1 | + 4i·2-s − 16·4-s + (43.5 − 35i)5-s + 17.4i·7-s − 64i·8-s + (140 + 174. i)10-s − 645.·11-s − 1.09e3i·13-s − 69.7·14-s + 256·16-s − 1.16e3i·17-s + 2.24e3·19-s + (−697. + 560i)20-s − 2.58e3i·22-s + 500i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (0.779 − 0.626i)5-s + 0.134i·7-s − 0.353i·8-s + (0.442 + 0.551i)10-s − 1.60·11-s − 1.80i·13-s − 0.0950·14-s + 0.250·16-s − 0.978i·17-s + 1.42·19-s + (−0.389 + 0.313i)20-s − 1.13i·22-s + 0.197i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.626 + 0.779i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.626 + 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.31255 - 0.629394i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.31255 - 0.629394i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-43.5 + 35i)T \) |
| good | 7 | \( 1 - 17.4iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 645.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.09e3iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 1.16e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 2.24e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 500iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 470.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.85e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 6.99e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 9.58e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 5.30e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 1.99e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 1.14e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 3.73e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.81e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.62e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 1.66e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.93e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 2.06e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.69e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 5.96e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 5.82e3iT - 8.58e9T^{2} \) |
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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
−13.26580446266511940123164532353, −12.26328386637947284642294839085, −10.49550551353545977207932504636, −9.644959810161730784999566748797, −8.337301901034010247843513820181, −7.40937087700404019310935425726, −5.57371754511329426895388020272, −5.17420518450467020485557048730, −2.84862985790611325952166676234, −0.59229486588608178435438139420,
1.73517065368450214618121847495, 3.04908629534306851877573308216, 4.80214290459624339511612084588, 6.23537907048040493572176812156, 7.68777205261882449604796187271, 9.205932870945034186401280892059, 10.17127011043058747011731325837, 11.00377265674596153288490431980, 12.15753082547558939220046953041, 13.47193284486884099358131168174
