L(s) = 1 | + (−0.5 + 0.866i)5-s + (−0.5 + 2.59i)7-s + (−1.5 − 2.59i)11-s + 4·13-s + (−2 + 3.46i)19-s + (−4 + 6.92i)23-s + (2 + 3.46i)25-s + 3·29-s + (−2.5 − 4.33i)31-s + (−2 − 1.73i)35-s + (−4 + 6.92i)37-s − 8·41-s − 6·43-s + (−5 + 8.66i)47-s + (−6.5 − 2.59i)49-s + ⋯ |
L(s) = 1 | + (−0.223 + 0.387i)5-s + (−0.188 + 0.981i)7-s + (−0.452 − 0.783i)11-s + 1.10·13-s + (−0.458 + 0.794i)19-s + (−0.834 + 1.44i)23-s + (0.400 + 0.692i)25-s + 0.557·29-s + (−0.449 − 0.777i)31-s + (−0.338 − 0.292i)35-s + (−0.657 + 1.13i)37-s − 1.24·41-s − 0.914·43-s + (−0.729 + 1.26i)47-s + (−0.928 − 0.371i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.083461160\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.083461160\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.5 - 2.59i)T \) |
good | 5 | \( 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2 - 3.46i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4 - 6.92i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4 - 6.92i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 + (5 - 8.66i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.5 - 7.79i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.5 - 4.33i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5 + 8.66i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3 - 5.19i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 10T + 71T^{2} \) |
| 73 | \( 1 + (1 + 1.73i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.5 + 9.52i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 7T + 83T^{2} \) |
| 89 | \( 1 + (9 - 15.5i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 17T + 97T^{2} \) |
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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
−10.22877178026562469538489130665, −9.377888471896250431845059888051, −8.418270096396682020024470256530, −7.990758084465177674660652182330, −6.69554715243966356567802592508, −5.94180364163104784273014017985, −5.24250366786821500690290514714, −3.74123121888927134381675693837, −3.06565535572265697567798398235, −1.66583108687459152086025706473,
0.49216883428349152510225449731, 2.04789090760621567865345414704, 3.51620374576969620171902952569, 4.37818740147083139828998627838, 5.18573818378977123964715791711, 6.59625619688329594845711747072, 6.97596438358924348736708629713, 8.302880065340558031601448467163, 8.585041065776713496977320329789, 9.929085178338103121391826670619