L(s) = 1 | + (0.866 + 0.5i)3-s + (−1.73 + 2i)7-s + (−1 − 1.73i)9-s + (−1.5 + 2.59i)11-s − 6i·13-s + (−4.33 − 2.5i)17-s + (0.5 + 0.866i)19-s + (−2.5 + 0.866i)21-s + (−6.06 + 3.5i)23-s − 5i·27-s − 2·29-s + (2.5 − 4.33i)31-s + (−2.59 + 1.5i)33-s + (−2.59 + 1.5i)37-s + (3 − 5.19i)39-s + ⋯ |
L(s) = 1 | + (0.499 + 0.288i)3-s + (−0.654 + 0.755i)7-s + (−0.333 − 0.577i)9-s + (−0.452 + 0.783i)11-s − 1.66i·13-s + (−1.05 − 0.606i)17-s + (0.114 + 0.198i)19-s + (−0.545 + 0.188i)21-s + (−1.26 + 0.729i)23-s − 0.962i·27-s − 0.371·29-s + (0.449 − 0.777i)31-s + (−0.452 + 0.261i)33-s + (−0.427 + 0.246i)37-s + (0.480 − 0.832i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.742 + 0.669i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.742 + 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3907040812\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3907040812\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.73 - 2i)T \) |
good | 3 | \( 1 + (-0.866 - 0.5i)T + (1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 + (4.33 + 2.5i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (6.06 - 3.5i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + (-2.5 + 4.33i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.59 - 1.5i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + (4.33 - 2.5i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.866 - 0.5i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-7.5 + 12.9i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.5 - 4.33i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (7.79 + 4.5i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (6.06 + 3.5i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + (-3.5 - 6.06i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−9.449341312307234054673209981501, −8.428129383748826318371661041744, −7.86128129450520321142914138477, −6.76874181712498439937589566431, −5.87592895374555893902510125540, −5.15933199347355835271774348098, −3.90822166245208246362551537234, −3.02976048155125047227069162442, −2.23570270255096852800178402951, −0.13475906297086189953520179102,
1.74988872079546523244696139985, 2.75169211778699393149820739983, 3.85941950528837757437702710607, 4.64353473214225688645223870650, 5.93926047337008311897925882340, 6.70057596776158568517647903846, 7.39801059638939267925580776571, 8.456818830885457868594111081994, 8.809589872147708968750312303238, 9.882870927344511970653481343681