L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s − 0.999·6-s + (−0.5 − 2.59i)7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (−2.5 + 4.33i)11-s + (−0.499 − 0.866i)12-s + (2 − 1.73i)14-s + (−0.5 − 0.866i)16-s + (−2 + 3.46i)17-s + (0.499 − 0.866i)18-s + (−4 − 6.92i)19-s + (2.5 + 0.866i)21-s − 5·22-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s − 0.408·6-s + (−0.188 − 0.981i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.753 + 1.30i)11-s + (−0.144 − 0.249i)12-s + (0.534 − 0.462i)14-s + (−0.125 − 0.216i)16-s + (−0.485 + 0.840i)17-s + (0.117 − 0.204i)18-s + (−0.917 − 1.58i)19-s + (0.545 + 0.188i)21-s − 1.06·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.5 + 2.59i)T \) |
good | 11 | \( 1 + (2.5 - 4.33i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + (2 - 3.46i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4 + 6.92i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 + (1.5 - 2.59i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2 + 3.46i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.5 - 7.79i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.5 + 9.52i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3 - 5.19i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + (-5 + 8.66i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.5 + 2.59i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 7T + 83T^{2} \) |
| 89 | \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 7T + 97T^{2} \) |
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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
−9.678689101341295452822122358106, −8.810438674498531201628486466349, −7.82401896633040635985045540793, −6.97977074030033089473429676998, −6.41007476745326135587539873599, −5.13071125691891994709698534688, −4.50595392829906558172204785000, −3.73006254594142138554368954655, −2.25711565454085477630407995337, 0,
1.78347108183832061810664122247, 2.78962558431298810407728049052, 3.78681309717082734740436612075, 5.21668572551171480318499492699, 5.76410473914077211476771835103, 6.51360396989113378177973354703, 7.86893710961468314708691687304, 8.493348126849116086106028590215, 9.434956655005391004339348037327