L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−2 − 3.46i)5-s − 0.999·8-s + (1.5 + 2.59i)9-s + (1.99 − 3.46i)10-s + (0.5 − 0.866i)11-s − 2·13-s + (−0.5 − 0.866i)16-s + (−2 + 3.46i)17-s + (−1.5 + 2.59i)18-s + (−3 − 5.19i)19-s + 3.99·20-s + 0.999·22-s + (−2 − 3.46i)23-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.894 − 1.54i)5-s − 0.353·8-s + (0.5 + 0.866i)9-s + (0.632 − 1.09i)10-s + (0.150 − 0.261i)11-s − 0.554·13-s + (−0.125 − 0.216i)16-s + (−0.485 + 0.840i)17-s + (−0.353 + 0.612i)18-s + (−0.688 − 1.19i)19-s + 0.894·20-s + 0.213·22-s + (−0.417 − 0.722i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3877161898\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3877161898\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
good | 3 | \( 1 + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (2 + 3.46i)T + (-2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + (2 - 3.46i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3 + 5.19i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (5 + 8.66i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 4T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + (-1 - 1.73i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (7 + 12.1i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6 + 10.3i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + (-2 + 3.46i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 14T + 97T^{2} \) |
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
−9.133009965674375628519104596176, −8.675465696989419823372271681782, −7.905878418643774710519691748612, −7.22680704622501398107131237885, −6.11518121962573745494109512909, −4.86054059875307597403338837200, −4.68253937992556196726861317861, −3.66503930610902484547606492912, −1.92525314806828388113684110865, −0.14636214555811217611461498834,
1.92952037325567972179124686277, 3.18515153612322293879083860020, 3.73793485382853907577824326267, 4.69961615649638562593921852332, 6.11054143141054975237765970775, 6.85635868816809327932155916651, 7.48860344856437729135364096388, 8.559011360012276137600174435808, 9.875038947519482512342610663379, 10.08000033787343492297607970989