L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (1.37 − 2.37i)5-s + (−1.37 − 2.26i)7-s − 0.999·8-s + (−1.37 − 2.37i)10-s + (0.5 + 0.866i)11-s + 5.49·13-s + (−2.64 + 0.0585i)14-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (4.01 − 6.96i)19-s − 2.74·20-s + 0.999·22-s + (−0.645 + 1.11i)23-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.614 − 1.06i)5-s + (−0.519 − 0.854i)7-s − 0.353·8-s + (−0.434 − 0.752i)10-s + (0.150 + 0.261i)11-s + 1.52·13-s + (−0.706 + 0.0156i)14-s + (−0.125 + 0.216i)16-s + (0.121 + 0.210i)17-s + (0.921 − 1.59i)19-s − 0.614·20-s + 0.213·22-s + (−0.134 + 0.232i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.820 + 0.572i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.820 + 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.058532104\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.058532104\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1.37 + 2.26i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
good | 5 | \( 1 + (-1.37 + 2.37i)T + (-2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 - 5.49T + 13T^{2} \) |
| 17 | \( 1 + (-0.5 - 0.866i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.01 + 6.96i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.645 - 1.11i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 4.54T + 29T^{2} \) |
| 31 | \( 1 + (2.54 + 4.40i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.01 + 3.49i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 5.54T + 41T^{2} \) |
| 43 | \( 1 + 4.03T + 43T^{2} \) |
| 47 | \( 1 + (4.64 - 8.04i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.74 - 4.75i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.76 + 8.25i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.829 + 1.43i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.77 + 3.06i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 2.54T + 71T^{2} \) |
| 73 | \( 1 + (-4.29 - 7.43i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (6.11 - 10.5i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 + (3.25 - 5.63i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 0.0872T + 97T^{2} \) |
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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
−9.384484096267960207605753746444, −8.769147961279720835789673623543, −7.66208817623988685097412047665, −6.62116726250057689452119837104, −5.77556296752932755866938475045, −4.94061394688219252873275905449, −4.04199420513766855944301247050, −3.20083201620226236588913291561, −1.70527046813312249582710211818, −0.78269220352060580761730316496,
1.79920726663430106721304094073, 3.21015468608521083762286225699, 3.58672772644776853918932395368, 5.21694306981016880856537558860, 6.01881266574770801683925561300, 6.34078796921556825565133797213, 7.28329325476683865915002168051, 8.303653197388050544965812772354, 8.962571927186156893540630160626, 9.918072700194274068655393528622