L(s) = 1 | + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (−1.73 − i)7-s + 0.999i·8-s + (3.46 − 2i)13-s + (−0.999 − 1.73i)14-s + (−0.5 + 0.866i)16-s + 6i·17-s + 7·19-s + 3.99·26-s − 1.99i·28-s + (3 − 5.19i)29-s + (5 + 8.66i)31-s + (−0.866 + 0.499i)32-s + (−3 + 5.19i)34-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.654 − 0.377i)7-s + 0.353i·8-s + (0.960 − 0.554i)13-s + (−0.267 − 0.462i)14-s + (−0.125 + 0.216i)16-s + 1.45i·17-s + 1.60·19-s + 0.784·26-s − 0.377i·28-s + (0.557 − 0.964i)29-s + (0.898 + 1.55i)31-s + (−0.153 + 0.0883i)32-s + (−0.514 + 0.891i)34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.687 - 0.726i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.687 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.375756961\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.375756961\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (1.73 + i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.46 + 2i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 - 7T + 19T^{2} \) |
| 23 | \( 1 + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-5 - 8.66i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + (-4.5 - 7.79i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.866 + 0.5i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.19 - 3i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 12iT - 53T^{2} \) |
| 59 | \( 1 + (-4.5 - 7.79i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (11.2 - 6.5i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 - iT - 73T^{2} \) |
| 79 | \( 1 + (-1 + 1.73i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (7.79 + 4.5i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 15T + 89T^{2} \) |
| 97 | \( 1 + (14.7 + 8.5i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.887084878846757428964970706644, −8.669410761440674063936295466714, −8.069014632590751208160826642105, −7.13119000039801879029430965183, −6.28564344775978034721416437487, −5.70941964054276276715791347915, −4.59748031213442751736779737114, −3.62067360650152616557976646609, −2.95314355167899729402243025374, −1.23058615344226498764497249373,
0.981968992972367962785342229785, 2.51776144510974271377720287776, 3.30045056090137089415652072363, 4.30450450655072953135092781700, 5.32486250723577321353382844823, 6.04350177303251539919452155249, 6.93151338559644522924258784023, 7.72343295008553425075211558402, 9.078639414934169132644247930193, 9.384600559969312409072901809031