L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s − 2.64·7-s − 0.999·8-s + (0.499 − 0.866i)10-s + (−0.322 + 0.559i)11-s − 4.64·13-s + (−1.32 − 2.29i)14-s + (−0.5 − 0.866i)16-s + (−3.64 + 6.31i)17-s + (−3.14 − 5.44i)19-s + 0.999·20-s − 0.645·22-s + (1.5 + 2.59i)23-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.223 − 0.387i)5-s − 0.999·7-s − 0.353·8-s + (0.158 − 0.273i)10-s + (−0.0973 + 0.168i)11-s − 1.28·13-s + (−0.353 − 0.612i)14-s + (−0.125 − 0.216i)16-s + (−0.884 + 1.53i)17-s + (−0.721 − 1.24i)19-s + 0.223·20-s − 0.137·22-s + (0.312 + 0.541i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0484169 - 0.206679i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0484169 - 0.206679i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + 2.64T \) |
good | 11 | \( 1 + (0.322 - 0.559i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 4.64T + 13T^{2} \) |
| 17 | \( 1 + (3.64 - 6.31i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.14 + 5.44i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.96 + 8.60i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 6.64T + 41T^{2} \) |
| 43 | \( 1 - 0.708T + 43T^{2} \) |
| 47 | \( 1 + (-5.79 - 10.0i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.14 + 8.91i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.645 + 1.11i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.354 + 0.613i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (7.64 - 13.2i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.354 + 0.613i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 4.70T + 83T^{2} \) |
| 89 | \( 1 + (7.29 + 12.6i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 4T + 97T^{2} \) |
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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
−11.08481338473126907646613090229, −10.12983084774957325796804859309, −9.157321080137198522791084071910, −8.537535185223491573670100459686, −7.30967207704915222522038721120, −6.73968559820279947960253948716, −5.68443977604347836822360161270, −4.66913807214883490127455936487, −3.74669549203566876781595131752, −2.39211587519885662535693945185,
0.094428370817041684284242128261, 2.32248839518092677922451884960, 3.18727320140620887071096029570, 4.33138524683159453624806246133, 5.36176574195248816607668149407, 6.55641637862877829259353852915, 7.21327315330696162195216130846, 8.518110473592264251612501592660, 9.514967281490672285717074963541, 10.15190191944451601182004353096