| L(s) = 1 | + (6.70 − 14.5i)2-s − 150.·3-s + (−166. − 194. i)4-s + (−1.00e3 + 2.18e3i)6-s − 2.62e3·7-s + (−3.94e3 + 1.10e3i)8-s + 1.60e4·9-s − 2.30e3i·11-s + (2.49e4 + 2.92e4i)12-s + 4.70e4i·13-s + (−1.76e4 + 3.81e4i)14-s + (−1.03e4 + 6.47e4i)16-s + 5.19e4i·17-s + (1.07e5 − 2.32e5i)18-s − 5.95e4i·19-s + ⋯ |
| L(s) = 1 | + (0.419 − 0.907i)2-s − 1.85·3-s + (−0.648 − 0.760i)4-s + (−0.777 + 1.68i)6-s − 1.09·7-s + (−0.962 + 0.270i)8-s + 2.43·9-s − 0.157i·11-s + (1.20 + 1.41i)12-s + 1.64i·13-s + (−0.458 + 0.993i)14-s + (−0.158 + 0.987i)16-s + 0.622i·17-s + (1.02 − 2.21i)18-s − 0.456i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.390 + 0.920i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.390 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.269569 - 0.407157i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.269569 - 0.407157i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-6.70 + 14.5i)T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 150.T + 6.56e3T^{2} \) |
| 7 | \( 1 + 2.62e3T + 5.76e6T^{2} \) |
| 11 | \( 1 + 2.30e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 4.70e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 - 5.19e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 5.95e4iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 7.75e4T + 7.83e10T^{2} \) |
| 29 | \( 1 + 9.02e5T + 5.00e11T^{2} \) |
| 31 | \( 1 + 3.40e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 5.84e5iT - 3.51e12T^{2} \) |
| 41 | \( 1 - 2.93e5T + 7.98e12T^{2} \) |
| 43 | \( 1 + 2.95e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + 5.03e6T + 2.38e13T^{2} \) |
| 53 | \( 1 + 7.54e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 - 8.82e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 1.08e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + 1.44e7T + 4.06e14T^{2} \) |
| 71 | \( 1 - 3.71e6iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 3.62e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 - 4.88e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 - 6.93e7T + 2.25e15T^{2} \) |
| 89 | \( 1 - 1.05e8T + 3.93e15T^{2} \) |
| 97 | \( 1 + 1.33e8iT - 7.83e15T^{2} \) |
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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
−11.75788741268828623183750511108, −11.19028895482060893797519872069, −10.12001383260698663706553392750, −9.272194036103235228233226055888, −6.78104056407577673898356406390, −6.07522312683394451066584450053, −4.87872035581509189744869700717, −3.76836423431151840417358888855, −1.71472558138084571142650636965, −0.30882368574640869742303059098,
0.56738732320194805404299429786, 3.46672043285477327331271125085, 4.99354904928802952634128260297, 5.77469793319882075316158171833, 6.64245118905479211958773372391, 7.65776789719990352657613335251, 9.535059328885756393373274273094, 10.51004149776319969822014935816, 11.80838122105402836413293354697, 12.75051674991670215953441038473
