L(s) = 1 | + (0.190 − 0.330i)2-s + (1.11 + 1.93i)3-s + (0.927 + 1.60i)4-s + (−1.11 + 1.93i)5-s + 0.854·6-s + 1.47·8-s + (−1 + 1.73i)9-s + (0.427 + 0.739i)10-s + (1.5 + 2.59i)11-s + (−2.07 + 3.59i)12-s + 13-s − 5.00·15-s + (−1.57 + 2.72i)16-s + (−3.73 − 6.47i)17-s + (0.381 + 0.661i)18-s + (1.5 − 2.59i)19-s + ⋯ |
L(s) = 1 | + (0.135 − 0.233i)2-s + (0.645 + 1.11i)3-s + (0.463 + 0.802i)4-s + (−0.499 + 0.866i)5-s + 0.348·6-s + 0.520·8-s + (−0.333 + 0.577i)9-s + (0.135 + 0.233i)10-s + (0.452 + 0.783i)11-s + (−0.598 + 1.03i)12-s + 0.277·13-s − 1.29·15-s + (−0.393 + 0.681i)16-s + (−0.906 − 1.56i)17-s + (0.0900 + 0.155i)18-s + (0.344 − 0.596i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.13789 + 1.71065i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.13789 + 1.71065i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + (-0.190 + 0.330i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-1.11 - 1.93i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (1.11 - 1.93i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (3.73 + 6.47i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.5 + 2.59i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.88 + 3.25i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 + (-2.5 - 4.33i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.35 + 7.54i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 4.47T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + (-0.736 + 1.27i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.736 + 1.27i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.73 - 6.47i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.5 + 2.59i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.5 - 2.59i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 8.94T + 71T^{2} \) |
| 73 | \( 1 + (-5.35 - 9.27i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.35 - 9.27i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (1.11 - 1.93i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 17.4T + 97T^{2} \) |
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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
−11.03941675167554705184430334832, −9.982805404547548157445837104087, −9.179818129592608039024636830643, −8.383531984492342104984833025764, −7.07142712545683308407459715000, −6.91975938625982184569873604364, −4.90336449348399530239699003410, −4.07349417103731306166888072790, −3.22908109928343167093342924606, −2.45378857554826320941896047546,
1.07959789309069802873356311898, 1.97072174160825553029468564769, 3.58190473073688495419791505823, 4.84654172195487196766523018450, 6.06978223505032457584478250949, 6.63557916723416195666476517112, 7.81633973978918382433892806065, 8.317530184685738677080348976393, 9.181774866275614461076697155143, 10.35033741356751167972868544244