L(s) = 1 | + (−2.13 − 1.23i)2-s + (1.47 + 0.901i)3-s + (2.04 + 3.54i)4-s + (1.98 − 3.43i)5-s + (−2.04 − 3.75i)6-s + (2.62 − 0.328i)7-s − 5.17i·8-s + (1.37 + 2.66i)9-s + (−8.46 + 4.88i)10-s + (−2.50 + 1.44i)11-s + (−0.170 + 7.09i)12-s − i·13-s + (−6.01 − 2.53i)14-s + (6.02 − 3.28i)15-s + (−2.28 + 3.96i)16-s + (2.85 + 4.94i)17-s + ⋯ |
L(s) = 1 | + (−1.51 − 0.872i)2-s + (0.853 + 0.520i)3-s + (1.02 + 1.77i)4-s + (0.885 − 1.53i)5-s + (−0.836 − 1.53i)6-s + (0.992 − 0.124i)7-s − 1.82i·8-s + (0.457 + 0.889i)9-s + (−2.67 + 1.54i)10-s + (−0.754 + 0.435i)11-s + (−0.0490 + 2.04i)12-s − 0.277i·13-s + (−1.60 − 0.678i)14-s + (1.55 − 0.848i)15-s + (−0.572 + 0.991i)16-s + (0.692 + 1.19i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.477 + 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.477 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.869158 - 0.517041i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.869158 - 0.517041i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.47 - 0.901i)T \) |
| 7 | \( 1 + (-2.62 + 0.328i)T \) |
| 13 | \( 1 + iT \) |
good | 2 | \( 1 + (2.13 + 1.23i)T + (1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (-1.98 + 3.43i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (2.50 - 1.44i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-2.85 - 4.94i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.00 + 1.15i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.664 + 0.383i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 5.11iT - 29T^{2} \) |
| 31 | \( 1 + (-0.564 + 0.326i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.28 + 2.22i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 2.44T + 41T^{2} \) |
| 43 | \( 1 + 8.81T + 43T^{2} \) |
| 47 | \( 1 + (1.84 - 3.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.17 + 2.98i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (0.198 + 0.343i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.42 - 1.97i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.12 - 1.95i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 5.63iT - 71T^{2} \) |
| 73 | \( 1 + (13.0 - 7.55i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.76 - 4.78i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 16.7T + 83T^{2} \) |
| 89 | \( 1 + (-3.59 + 6.22i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 2.08iT - 97T^{2} \) |
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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
−11.44635156649361429769187775141, −10.19497821215376410084426155389, −10.00124883381183126647323215013, −8.820583562356778800620245639333, −8.361251506608670223758606807114, −7.69183421841346647177251864315, −5.41593739802820149189927139737, −4.24369513123788133483605600429, −2.36299904297735565662814494525, −1.43934396782571110045283178500,
1.72810737299691503285058903488, 2.90314700767558762634225365476, 5.55676992505910811200273899154, 6.63987521587885570357290169849, 7.35108373603708598776008176873, 8.087703443102198221989062117893, 9.047111110793480857006705345535, 9.987049047314646338440345777340, 10.63161790708496582618734883805, 11.66232889372837599532452695836