L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.224 − 0.389i)7-s − 0.999·8-s + (−2.44 − 4.24i)11-s + (−0.224 + 0.389i)13-s + (0.224 − 0.389i)14-s + (−0.5 − 0.866i)16-s + 4.89·17-s + 7.44·19-s + (2.44 − 4.24i)22-s + (1.22 − 2.12i)23-s − 0.449·26-s + 0.449·28-s + (−1.22 − 2.12i)29-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.0849 − 0.147i)7-s − 0.353·8-s + (−0.738 − 1.27i)11-s + (−0.0623 + 0.107i)13-s + (0.0600 − 0.104i)14-s + (−0.125 − 0.216i)16-s + 1.18·17-s + 1.70·19-s + (0.522 − 0.904i)22-s + (0.255 − 0.442i)23-s − 0.0881·26-s + 0.0849·28-s + (−0.227 − 0.393i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.263i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 - 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.901182167\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.901182167\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 7 | \( 1 + (0.224 + 0.389i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.44 + 4.24i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.224 - 0.389i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 4.89T + 17T^{2} \) |
| 19 | \( 1 - 7.44T + 19T^{2} \) |
| 23 | \( 1 + (-1.22 + 2.12i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.22 + 2.12i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (2.22 - 3.85i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 + (-4.5 + 7.79i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.27 - 2.20i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.44 - 9.43i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 3.55T + 53T^{2} \) |
| 59 | \( 1 + (2.72 - 4.71i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.174 + 0.301i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 13.3T + 71T^{2} \) |
| 73 | \( 1 - T + 73T^{2} \) |
| 79 | \( 1 + (8.34 + 14.4i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.72 - 4.71i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 9T + 89T^{2} \) |
| 97 | \( 1 + (-4.39 - 7.61i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.495818872088517704225229187408, −8.757765196142176455463614518187, −7.67121429884988859285594755285, −7.49348240492192730613841757152, −6.07523406253942200591376850841, −5.65319169625061370369477086694, −4.71382872234271675805605402611, −3.50878746298821346101143924832, −2.82395994427861720506980348040, −0.856592559022417067197807360179,
1.16690437668049483212765498259, 2.47083233220812595710562016841, 3.32701819704112125281820913082, 4.45941557643296133835659790203, 5.29873464672879007993362095721, 5.96798187137750553835905045026, 7.48024650662133991954777859034, 7.61726737988816230960351254218, 9.077645072178665963603277411097, 9.778869169254062671394200024680