L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (2.5 − 0.866i)7-s − 0.999·8-s + (−0.5 − 0.866i)11-s + 5·13-s + (0.500 − 2.59i)14-s + (−0.5 + 0.866i)16-s + (3 + 5.19i)17-s + (−1 + 1.73i)19-s − 0.999·22-s + (3 − 5.19i)23-s + (2.5 + 4.33i)25-s + (2.5 − 4.33i)26-s + (−1.99 − 1.73i)28-s − 3·29-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.944 − 0.327i)7-s − 0.353·8-s + (−0.150 − 0.261i)11-s + 1.38·13-s + (0.133 − 0.694i)14-s + (−0.125 + 0.216i)16-s + (0.727 + 1.26i)17-s + (−0.229 + 0.397i)19-s − 0.213·22-s + (0.625 − 1.08i)23-s + (0.5 + 0.866i)25-s + (0.490 − 0.849i)26-s + (−0.377 − 0.327i)28-s − 0.557·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.324242512\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.324242512\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.5 + 0.866i)T \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
good | 5 | \( 1 + (-2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 - 5T + 13T^{2} \) |
| 17 | \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + (4 + 6.92i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (6 + 10.3i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.5 - 11.2i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + (-5 - 8.66i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 13T + 97T^{2} \) |
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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
−9.542029117643362321591924420731, −8.451100725055314291334004970982, −8.147951823831937719370681748093, −6.89413141030597965906877147854, −5.89666689105492550792566417797, −5.21335681472076505683043377906, −4.05808537982217627440062105224, −3.50870654016409637107085711208, −2.04283183714581064365589456438, −1.06698112310847881631634707623,
1.27505483400045629054360020248, 2.76390606311101058541979669854, 3.82410473994110530795284801153, 4.91270848811148986621272277109, 5.44033758188113459763450083063, 6.41716003807162379629165287649, 7.36251084211016189474548588857, 7.961537165579687276118566942626, 8.889022219218410515695836530303, 9.379555336746172130787091316693