| L(s) = 1 | + (−1.49 − 1.32i)2-s + (−1.96 − 2.26i)3-s + (0.466 + 3.97i)4-s + (−5.84 + 3.37i)5-s + (−0.0664 + 5.99i)6-s + (3.50 + 2.02i)7-s + (4.58 − 6.55i)8-s + (−1.24 + 8.91i)9-s + (13.2 + 2.72i)10-s + (−4.13 + 7.16i)11-s + (8.07 − 8.87i)12-s + (−7.66 + 4.42i)13-s + (−2.55 − 7.68i)14-s + (19.1 + 6.58i)15-s + (−15.5 + 3.70i)16-s − 28.6·17-s + ⋯ |
| L(s) = 1 | + (−0.747 − 0.664i)2-s + (−0.656 − 0.754i)3-s + (0.116 + 0.993i)4-s + (−1.16 + 0.675i)5-s + (−0.0110 + 0.999i)6-s + (0.501 + 0.289i)7-s + (0.572 − 0.819i)8-s + (−0.138 + 0.990i)9-s + (1.32 + 0.272i)10-s + (−0.376 + 0.651i)11-s + (0.672 − 0.739i)12-s + (−0.589 + 0.340i)13-s + (−0.182 − 0.549i)14-s + (1.27 + 0.439i)15-s + (−0.972 + 0.231i)16-s − 1.68·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.123 - 0.992i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.123 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.137386 + 0.155500i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.137386 + 0.155500i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.49 + 1.32i)T \) |
| 3 | \( 1 + (1.96 + 2.26i)T \) |
| good | 5 | \( 1 + (5.84 - 3.37i)T + (12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (-3.50 - 2.02i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (4.13 - 7.16i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (7.66 - 4.42i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 28.6T + 289T^{2} \) |
| 19 | \( 1 + 7.93T + 361T^{2} \) |
| 23 | \( 1 + (-25.3 + 14.6i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (15.7 + 9.08i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (40.8 - 23.5i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 13.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-31.0 - 53.7i)T + (-840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-26.5 + 46.0i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-12.8 - 7.43i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 - 100. iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (11.4 + 19.9i)T + (-1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-51.3 - 29.6i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-10.3 - 17.9i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 54.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 27.3T + 5.32e3T^{2} \) |
| 79 | \( 1 + (62.9 + 36.3i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-4.34 + 7.53i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 28.2T + 7.92e3T^{2} \) |
| 97 | \( 1 + (0.200 - 0.348i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−14.81697402725322180686110442162, −13.07741561879995511573359980819, −12.21599136838725666388775052580, −11.23062693432833325915062690650, −10.74930752278292122537063012712, −8.868685748858270217992289438306, −7.58726187843746021395690777109, −6.91353478005696006268174625628, −4.50346537978192658806022390563, −2.33361174424102416204732382214,
0.23917326317863035848333957878, 4.32553370412109093608279647869, 5.42739770256330839393748821439, 7.13376413501843250381502606568, 8.369549148736174563669989362367, 9.334206263057622663459449955910, 10.94706556278486524510417607892, 11.29516696501429556422803455030, 12.93335255980791742993172491178, 14.68990344161862552583693291307
