L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (−1.82 − 3.15i)5-s + (1.32 + 2.29i)7-s + 0.999·8-s + (−1.82 + 3.15i)10-s + (0.322 − 0.559i)11-s + 13-s + (1.32 − 2.29i)14-s + (−0.5 − 0.866i)16-s + (−3.32 + 5.75i)17-s + (−2.5 − 4.33i)19-s + 3.64·20-s − 0.645·22-s + (−1.17 − 2.03i)23-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.815 − 1.41i)5-s + (0.499 + 0.866i)7-s + 0.353·8-s + (−0.576 + 0.998i)10-s + (0.0973 − 0.168i)11-s + 0.277·13-s + (0.353 − 0.612i)14-s + (−0.125 − 0.216i)16-s + (−0.805 + 1.39i)17-s + (−0.573 − 0.993i)19-s + 0.815·20-s − 0.137·22-s + (−0.245 − 0.425i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2775609447\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2775609447\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (-1.32 - 2.29i)T \) |
| 13 | \( 1 - T \) |
good | 5 | \( 1 + (1.82 + 3.15i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.322 + 0.559i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (3.32 - 5.75i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.5 + 4.33i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.17 + 2.03i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 4.29T + 29T^{2} \) |
| 31 | \( 1 + (1.64 - 2.85i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.82 + 4.88i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 2.35T + 41T^{2} \) |
| 43 | \( 1 + 5.29T + 43T^{2} \) |
| 47 | \( 1 + (-1.5 - 2.59i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.5 - 2.59i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.96 - 6.87i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.96 - 10.3i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.79 - 6.56i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 16.2T + 71T^{2} \) |
| 73 | \( 1 + (-6.82 + 11.8i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-5 - 8.66i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 + (-8.46 - 14.6i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 0.937T + 97T^{2} \) |
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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
−9.097603598975220011355775566379, −8.899525553049583262740067327998, −8.328842694756239380556452124294, −7.55040138354461976354855034778, −6.26286108072198574846384111796, −5.25636479859120463839027151465, −4.44849621464829863566020835816, −3.78333597354226173200310627106, −2.33541034715053379919063573575, −1.32305886670911504502077320400,
0.12496992900979926841914929406, 1.93377480380183226901149110161, 3.34510457207087140901389637118, 4.09494275655900844671445158349, 5.06884078556900675715565155716, 6.34842504328708911622679400679, 6.88147249201879466069125056572, 7.62182470498022745892343404528, 8.022303984563423528162673010792, 9.144860580889699363349714422004