L(s) = 1 | + (1.38 + 0.295i)2-s + 1.66i·3-s + (1.82 + 0.817i)4-s + (−2.22 − 0.220i)5-s + (−0.491 + 2.29i)6-s + (2.28 + 1.67i)8-s + 0.236·9-s + (−3.01 − 0.962i)10-s + 4.81i·11-s + (−1.35 + 3.03i)12-s + 2.14·13-s + (0.366 − 3.69i)15-s + (2.66 + 2.98i)16-s − 5.02·17-s + (0.326 + 0.0698i)18-s + 1.36·19-s + ⋯ |
L(s) = 1 | + (0.977 + 0.209i)2-s + 0.959i·3-s + (0.912 + 0.408i)4-s + (−0.995 − 0.0986i)5-s + (−0.200 + 0.938i)6-s + (0.807 + 0.590i)8-s + 0.0787·9-s + (−0.952 − 0.304i)10-s + 1.45i·11-s + (−0.392 + 0.875i)12-s + 0.594·13-s + (0.0946 − 0.955i)15-s + (0.665 + 0.746i)16-s − 1.21·17-s + (0.0770 + 0.0164i)18-s + 0.312·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.588 - 0.808i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.588 - 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.10016 + 2.16219i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.10016 + 2.16219i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.38 - 0.295i)T \) |
| 5 | \( 1 + (2.22 + 0.220i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 1.66iT - 3T^{2} \) |
| 11 | \( 1 - 4.81iT - 11T^{2} \) |
| 13 | \( 1 - 2.14T + 13T^{2} \) |
| 17 | \( 1 + 5.02T + 17T^{2} \) |
| 19 | \( 1 - 1.36T + 19T^{2} \) |
| 23 | \( 1 + 5.18T + 23T^{2} \) |
| 29 | \( 1 + 6.43T + 29T^{2} \) |
| 31 | \( 1 - 4.62T + 31T^{2} \) |
| 37 | \( 1 + 9.82iT - 37T^{2} \) |
| 41 | \( 1 - 4.71iT - 41T^{2} \) |
| 43 | \( 1 + 0.141T + 43T^{2} \) |
| 47 | \( 1 - 2.55iT - 47T^{2} \) |
| 53 | \( 1 - 4.84iT - 53T^{2} \) |
| 59 | \( 1 - 14.1T + 59T^{2} \) |
| 61 | \( 1 - 10.1iT - 61T^{2} \) |
| 67 | \( 1 - 9.64T + 67T^{2} \) |
| 71 | \( 1 + 9.58iT - 71T^{2} \) |
| 73 | \( 1 - 1.67T + 73T^{2} \) |
| 79 | \( 1 + 11.8iT - 79T^{2} \) |
| 83 | \( 1 - 0.811iT - 83T^{2} \) |
| 89 | \( 1 + 16.0iT - 89T^{2} \) |
| 97 | \( 1 + 1.76T + 97T^{2} \) |
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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
−10.47017606855606258260373068262, −9.565891052752660472356206097250, −8.563957804455451967723695319483, −7.52054126210171622867196079696, −6.98050902497646584582941350981, −5.79503385184719709603323742621, −4.60464491971378821969996693502, −4.30218179851763191247506378564, −3.49696458549048418983314751735, −2.05024142056754997725837740375,
0.825655181589445869522065638230, 2.23067182868036723679326509566, 3.46535346885381107311563917226, 4.15097351485718015513071422028, 5.39555956988354582780557726910, 6.44434882507246218409984017713, 6.87394429436870530795189206374, 7.979431529234525053867201693595, 8.462651645021202855959487143983, 9.942480999667496945428290111568