L(s) = 1 | + (−0.309 + 0.951i)2-s + (−0.809 − 0.587i)4-s + (−0.411 + 0.133i)5-s + (−0.587 + 0.809i)7-s + (0.809 − 0.587i)8-s − 0.432i·10-s + (0.137 − 3.31i)11-s + (−1.34 − 0.436i)13-s + (−0.587 − 0.809i)14-s + (0.309 + 0.951i)16-s + (0.223 + 0.688i)17-s + (3.54 + 4.87i)19-s + (0.411 + 0.133i)20-s + (3.10 + 1.15i)22-s + 1.67i·23-s + ⋯ |
L(s) = 1 | + (−0.218 + 0.672i)2-s + (−0.404 − 0.293i)4-s + (−0.184 + 0.0598i)5-s + (−0.222 + 0.305i)7-s + (0.286 − 0.207i)8-s − 0.136i·10-s + (0.0414 − 0.999i)11-s + (−0.372 − 0.121i)13-s + (−0.157 − 0.216i)14-s + (0.0772 + 0.237i)16-s + (0.0542 + 0.166i)17-s + (0.812 + 1.11i)19-s + (0.0920 + 0.0299i)20-s + (0.662 + 0.246i)22-s + 0.348i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.778 - 0.627i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.778 - 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8678834838\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8678834838\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.309 - 0.951i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.587 - 0.809i)T \) |
| 11 | \( 1 + (-0.137 + 3.31i)T \) |
good | 5 | \( 1 + (0.411 - 0.133i)T + (4.04 - 2.93i)T^{2} \) |
| 13 | \( 1 + (1.34 + 0.436i)T + (10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-0.223 - 0.688i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-3.54 - 4.87i)T + (-5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 - 1.67iT - 23T^{2} \) |
| 29 | \( 1 + (0.367 + 0.266i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (1.99 - 6.15i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-6.85 - 4.98i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (1.35 - 0.985i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 5.52iT - 43T^{2} \) |
| 47 | \( 1 + (2.88 + 3.97i)T + (-14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (0.253 + 0.0824i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (3.56 - 4.90i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-0.756 + 0.245i)T + (49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + 0.680T + 67T^{2} \) |
| 71 | \( 1 + (2.38 - 0.774i)T + (57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (3.78 - 5.20i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (5.90 + 1.92i)T + (63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (1.83 + 5.64i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 - 8.71iT - 89T^{2} \) |
| 97 | \( 1 + (5.31 - 16.3i)T + (-78.4 - 57.0i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.737545569626063378892213818250, −9.030390755942253235443429089708, −8.120447178592140385667244687007, −7.63120067572277628294748570454, −6.57873542480137205307478108797, −5.81313722554268249944659834743, −5.15102814394013447188341093525, −3.87365943476986200683836294713, −3.00608725563344206500052700383, −1.33936749953110952684946813178,
0.40371484642263954426690998955, 1.95445257754415349398617112876, 2.92973714045103573197875220066, 4.11971070731648641957511357769, 4.73757428885240699301908835170, 5.88922714490599444672720633876, 7.11943206035059228804400083348, 7.56129766941121379213448360909, 8.603589784486534141720180978162, 9.642701304559829866394129295006