L(s) = 1 | + (−1 − 1.73i)3-s + (−0.5 + 0.866i)5-s + (−0.499 + 0.866i)9-s − 2·13-s + 1.99·15-s + (−3 − 5.19i)17-s + (−2 + 3.46i)19-s + (−3 + 5.19i)23-s + (−0.499 − 0.866i)25-s − 4.00·27-s + 6·29-s + (−2 − 3.46i)31-s + (−1 + 1.73i)37-s + (2 + 3.46i)39-s − 6·41-s + ⋯ |
L(s) = 1 | + (−0.577 − 0.999i)3-s + (−0.223 + 0.387i)5-s + (−0.166 + 0.288i)9-s − 0.554·13-s + 0.516·15-s + (−0.727 − 1.26i)17-s + (−0.458 + 0.794i)19-s + (−0.625 + 1.08i)23-s + (−0.0999 − 0.173i)25-s − 0.769·27-s + 1.11·29-s + (−0.359 − 0.622i)31-s + (−0.164 + 0.284i)37-s + (0.320 + 0.554i)39-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (1 + 1.73i)T + (-1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2 - 3.46i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 10T + 43T^{2} \) |
| 47 | \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 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 | \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1 + 1.73i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + (-1 - 1.73i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−9.609312979669962421036077568725, −8.469029290848132533946342079075, −7.55063861849093784405400132968, −6.96309481843633676754822466877, −6.22388277960908917403967252649, −5.27782986439097672900272675997, −4.12663907617630997213205759291, −2.81707654877198533725577674054, −1.59507968108844440572611551061, 0,
2.05404859426744015397564861568, 3.59689222064108361214940628665, 4.56146916215330830942690266943, 5.01832465489147199907528245066, 6.18254081660402112783335453292, 7.00697775460501886819900456835, 8.346358474554124649413682515239, 8.765618259406640218291553476730, 10.03965015248128628912106807092