L(s) = 1 | + (−1.63 − 1.15i)2-s + (2.67 + 1.36i)3-s + (1.32 + 3.77i)4-s + (3.07 − 5.32i)5-s + (−2.78 − 5.31i)6-s + (−0.511 + 0.295i)7-s + (2.20 − 7.68i)8-s + (5.27 + 7.29i)9-s + (−11.1 + 5.12i)10-s + (−15.1 + 8.72i)11-s + (−1.61 + 11.8i)12-s + (−0.892 + 1.54i)13-s + (1.17 + 0.110i)14-s + (15.4 − 10.0i)15-s + (−12.5 + 9.98i)16-s − 16.9·17-s + ⋯ |
L(s) = 1 | + (−0.815 − 0.578i)2-s + (0.890 + 0.454i)3-s + (0.330 + 0.943i)4-s + (0.614 − 1.06i)5-s + (−0.463 − 0.886i)6-s + (−0.0730 + 0.0421i)7-s + (0.276 − 0.961i)8-s + (0.586 + 0.810i)9-s + (−1.11 + 0.512i)10-s + (−1.37 + 0.793i)11-s + (−0.134 + 0.990i)12-s + (−0.0686 + 0.118i)13-s + (0.0840 + 0.00785i)14-s + (1.03 − 0.668i)15-s + (−0.781 + 0.624i)16-s − 0.995·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 + 0.419i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.907 + 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.930714 - 0.204565i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.930714 - 0.204565i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.63 + 1.15i)T \) |
| 3 | \( 1 + (-2.67 - 1.36i)T \) |
good | 5 | \( 1 + (-3.07 + 5.32i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (0.511 - 0.295i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (15.1 - 8.72i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (0.892 - 1.54i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 16.9T + 289T^{2} \) |
| 19 | \( 1 + 19.5iT - 361T^{2} \) |
| 23 | \( 1 + (-6.86 - 3.96i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-3.17 - 5.49i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (27.6 + 15.9i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 58.2T + 1.36e3T^{2} \) |
| 41 | \( 1 + (2.66 - 4.62i)T + (-840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-33.9 + 19.5i)T + (924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-9.64 + 5.56i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 - 35.8T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-20.8 - 12.0i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (37.9 + 65.7i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-31.8 - 18.3i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 87.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 60.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + (32.1 - 18.5i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-66.0 + 38.1i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 27.5T + 7.92e3T^{2} \) |
| 97 | \( 1 + (-13.0 - 22.6i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−16.23035015908640897144154056890, −15.31500623576593659524529528409, −13.32157233555041307695194659963, −12.82793412733246489958346771427, −10.87806137120566756859799447059, −9.616125019791540642781013332349, −8.880319984339331217379974398818, −7.55056924767514320459927913229, −4.69096108301895870049865297326, −2.35306030444287242496244825486,
2.55650648891392223678608191777, 6.03719835621190826860148789830, 7.32274525764752044460570805583, 8.495979649830652217109152886895, 9.928676991465239792193085974513, 10.92710551674622477819092656720, 13.17646113251987227840619823148, 14.20978125527106688332039353996, 15.08876058633547078987813408011, 16.25172751716111494672437930464