| 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
−9.882870927344511970653481343681, −8.809589872147708968750312303238, −8.456818830885457868594111081994, −7.39801059638939267925580776571, −6.70057596776158568517647903846, −5.93926047337008311897925882340, −4.64353473214225688645223870650, −3.85941950528837757437702710607, −2.75169211778699393149820739983, −1.74988872079546523244696139985,
0.13475906297086189953520179102, 2.23570270255096852800178402951, 3.02976048155125047227069162442, 3.90822166245208246362551537234, 5.15933199347355835271774348098, 5.87592895374555893902510125540, 6.76874181712498439937589566431, 7.86128129450520321142914138477, 8.428129383748826318371661041744, 9.449341312307234054673209981501
