L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s − 5-s + (0.499 + 0.866i)6-s + (2.5 + 4.33i)7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)10-s + (1.5 − 2.59i)11-s + 0.999·12-s + (−2.5 + 2.59i)13-s + 5·14-s + (0.5 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (4 + 6.92i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s − 0.447·5-s + (0.204 + 0.353i)6-s + (0.944 + 1.63i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.158 + 0.273i)10-s + (0.452 − 0.783i)11-s + 0.288·12-s + (−0.693 + 0.720i)13-s + 1.33·14-s + (0.129 − 0.223i)15-s + (−0.125 + 0.216i)16-s + (0.970 + 1.68i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.37206 + 0.377090i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.37206 + 0.377090i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + (2.5 - 2.59i)T \) |
good | 7 | \( 1 + (-2.5 - 4.33i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-4 - 6.92i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2 + 3.46i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + (-3.5 + 6.06i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3 - 5.19i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3 + 5.19i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 3T + 47T^{2} \) |
| 53 | \( 1 - T + 53T^{2} \) |
| 59 | \( 1 + (6 + 10.3i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4 - 6.92i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (1 + 1.73i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 2T + 79T^{2} \) |
| 83 | \( 1 - 8T + 83T^{2} \) |
| 89 | \( 1 + (-5.5 + 9.52i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-48.5 + 84.0i)T^{2} \) |
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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
−11.58781777065658408086541356225, −10.71019029806201543855336521509, −9.655925829411925947797474293582, −8.711213230850735519341622482761, −8.017329282219072338940324245706, −6.17842300625523799904966333156, −5.47887409253854378312717854064, −4.43515130368606390189860790858, −3.28864928362008128579029329385, −1.80402139087837853590352056605,
0.982624931644370404091049194855, 3.20538254383081894545876644311, 4.66754358556719115491956771686, 5.12793525022361651353586049475, 6.91521316024092169532631252059, 7.37674679339913567309724890579, 7.88127892647357733272462422502, 9.383577281595628268269351009975, 10.41513631541863398691472256687, 11.49609631662168158332695502945