L(s) = 1 | + (0.5 − 0.866i)5-s + (0.835 + 2.51i)7-s + (−3.11 − 5.39i)11-s − 3.77·13-s + (−0.313 − 0.542i)17-s + (−0.206 + 0.357i)19-s + (−2.04 + 3.53i)23-s + (−0.499 − 0.866i)25-s + 4.12·29-s + (4.32 + 7.49i)31-s + (2.59 + 0.532i)35-s + (−3.59 + 6.22i)37-s − 4.88·41-s − 10.6·43-s + (−5.49 + 9.52i)47-s + ⋯ |
L(s) = 1 | + (0.223 − 0.387i)5-s + (0.315 + 0.948i)7-s + (−0.938 − 1.62i)11-s − 1.04·13-s + (−0.0759 − 0.131i)17-s + (−0.0474 + 0.0821i)19-s + (−0.425 + 0.737i)23-s + (−0.0999 − 0.173i)25-s + 0.766·29-s + (0.777 + 1.34i)31-s + (0.438 + 0.0899i)35-s + (−0.590 + 1.02i)37-s − 0.763·41-s − 1.62·43-s + (−0.802 + 1.38i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.788 - 0.615i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4517135512\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4517135512\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (-0.835 - 2.51i)T \) |
good | 11 | \( 1 + (3.11 + 5.39i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 3.77T + 13T^{2} \) |
| 17 | \( 1 + (0.313 + 0.542i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.206 - 0.357i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.04 - 3.53i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 4.12T + 29T^{2} \) |
| 31 | \( 1 + (-4.32 - 7.49i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.59 - 6.22i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 4.88T + 41T^{2} \) |
| 43 | \( 1 + 10.6T + 43T^{2} \) |
| 47 | \( 1 + (5.49 - 9.52i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.271 - 0.470i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.16 - 7.21i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.963 + 1.66i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.50 + 6.07i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 + (2.58 + 4.47i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.57 - 6.19i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 10.0T + 83T^{2} \) |
| 89 | \( 1 + (-1.60 + 2.78i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 13.7T + 97T^{2} \) |
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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
−9.097344882604655199770324476643, −8.262989632981353746224322883185, −8.111625877318152304655269268182, −6.81701752919050881923100729265, −6.00606935149957435636334285470, −5.19974672613106551845836158248, −4.82725405705475356865534531200, −3.27332544926560389335648829279, −2.67769014197661838164826881530, −1.44730267266190528269439831205,
0.14006408836172163893677899466, 1.89049520780728281273025617133, 2.56792915062809416001204961660, 3.86869475425694790695853941139, 4.71887405507961198147933898769, 5.22857810608549133020265356556, 6.59089149011807981171007902948, 7.05678547385874437420006294356, 7.76871159633980044785389148755, 8.411231077660840635775047924187