L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.866 + 0.5i)3-s + (−0.499 − 0.866i)4-s + (1.03 + 1.98i)5-s − 0.999i·6-s + (−3.17 + 1.83i)7-s + 0.999·8-s + (0.499 − 0.866i)9-s + (−2.23 − 0.0953i)10-s + 4.74·11-s + (0.866 + 0.499i)12-s + (−1.59 − 2.76i)13-s − 3.66i·14-s + (−1.88 − 1.19i)15-s + (−0.5 + 0.866i)16-s + (−2.99 + 5.18i)17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.499 + 0.288i)3-s + (−0.249 − 0.433i)4-s + (0.462 + 0.886i)5-s − 0.408i·6-s + (−1.19 + 0.692i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (−0.706 − 0.0301i)10-s + 1.43·11-s + (0.249 + 0.144i)12-s + (−0.442 − 0.766i)13-s − 0.979i·14-s + (−0.487 − 0.309i)15-s + (−0.125 + 0.216i)16-s + (−0.725 + 1.25i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.592 + 0.805i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.592 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3719555197\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3719555197\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.866 - 0.5i)T \) |
| 5 | \( 1 + (-1.03 - 1.98i)T \) |
| 37 | \( 1 + (-5.81 - 1.77i)T \) |
good | 7 | \( 1 + (3.17 - 1.83i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 - 4.74T + 11T^{2} \) |
| 13 | \( 1 + (1.59 + 2.76i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (2.99 - 5.18i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4.87 - 2.81i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 2.02T + 23T^{2} \) |
| 29 | \( 1 - 3.80iT - 29T^{2} \) |
| 31 | \( 1 + 6.38iT - 31T^{2} \) |
| 41 | \( 1 + (-0.175 - 0.304i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 - 3.71T + 43T^{2} \) |
| 47 | \( 1 + 10.1iT - 47T^{2} \) |
| 53 | \( 1 + (6.59 + 3.80i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (8.74 + 5.04i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (13.3 - 7.68i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.485 - 0.280i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (6.31 + 10.9i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 2.67iT - 73T^{2} \) |
| 79 | \( 1 + (-1.65 + 0.955i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (4.66 + 2.69i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-10.8 - 6.23i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 3.07T + 97T^{2} \) |
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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
−10.28376535416531364120648640217, −9.507111460127144281991980299208, −8.962657104319150777676455100267, −7.84448050924986073453956389695, −6.66042338644778824520775199957, −6.22631397022096655377407871408, −5.82373639014125778828031017820, −4.30758298104832001800661154765, −3.34085114096148573732986779732, −1.95313914746614534459293125332,
0.19960743092342717418822633925, 1.40271989087898487225552784294, 2.67709143711363060938168763333, 4.18337362288211047111088821072, 4.62025279971909773256048048212, 6.20339276795285985162093580080, 6.65327730387605298405140938087, 7.59586552843361350721773032850, 9.057980473106224861084679496030, 9.207827362371725825095778888236