L(s) = 1 | + (0.597 − 0.802i)2-s + (0.893 − 0.448i)3-s + (−0.286 − 0.957i)4-s + (−0.835 − 0.549i)5-s + (0.173 − 0.984i)6-s + (0.0798 + 0.0845i)7-s + (−0.939 − 0.342i)8-s + (0.597 − 0.802i)9-s + (−0.939 + 0.342i)10-s + (−0.686 − 0.727i)12-s + (0.115 − 0.0135i)14-s + (−0.993 − 0.116i)15-s + (−0.835 + 0.549i)16-s + (−0.286 − 0.957i)18-s + (−0.286 + 0.957i)20-s + (0.109 + 0.0397i)21-s + ⋯ |
L(s) = 1 | + (0.597 − 0.802i)2-s + (0.893 − 0.448i)3-s + (−0.286 − 0.957i)4-s + (−0.835 − 0.549i)5-s + (0.173 − 0.984i)6-s + (0.0798 + 0.0845i)7-s + (−0.939 − 0.342i)8-s + (0.597 − 0.802i)9-s + (−0.939 + 0.342i)10-s + (−0.686 − 0.727i)12-s + (0.115 − 0.0135i)14-s + (−0.993 − 0.116i)15-s + (−0.835 + 0.549i)16-s + (−0.286 − 0.957i)18-s + (−0.286 + 0.957i)20-s + (0.109 + 0.0397i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.654496495\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.654496495\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.597 + 0.802i)T \) |
| 3 | \( 1 + (-0.893 + 0.448i)T \) |
| 5 | \( 1 + (0.835 + 0.549i)T \) |
good | 7 | \( 1 + (-0.0798 - 0.0845i)T + (-0.0581 + 0.998i)T^{2} \) |
| 11 | \( 1 + (-0.597 - 0.802i)T^{2} \) |
| 13 | \( 1 + (0.686 + 0.727i)T^{2} \) |
| 17 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 19 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 23 | \( 1 + (-0.941 + 0.998i)T + (-0.0581 - 0.998i)T^{2} \) |
| 29 | \( 1 + (1.93 + 0.225i)T + (0.973 + 0.230i)T^{2} \) |
| 31 | \( 1 + (-0.893 - 0.448i)T^{2} \) |
| 37 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 41 | \( 1 + (-0.473 - 0.635i)T + (-0.286 + 0.957i)T^{2} \) |
| 43 | \( 1 + (-0.109 - 1.87i)T + (-0.993 + 0.116i)T^{2} \) |
| 47 | \( 1 + (-1.16 + 0.275i)T + (0.893 - 0.448i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.597 + 0.802i)T^{2} \) |
| 61 | \( 1 + (0.512 - 1.71i)T + (-0.835 - 0.549i)T^{2} \) |
| 67 | \( 1 + (-1.97 + 0.230i)T + (0.973 - 0.230i)T^{2} \) |
| 71 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 73 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 79 | \( 1 + (0.286 + 0.957i)T^{2} \) |
| 83 | \( 1 + (-0.473 + 0.635i)T + (-0.286 - 0.957i)T^{2} \) |
| 89 | \( 1 + (1.12 + 0.408i)T + (0.766 + 0.642i)T^{2} \) |
| 97 | \( 1 + (-0.396 + 0.918i)T^{2} \) |
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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
−9.176113244196944438818881859098, −8.678185574663712512080032928065, −7.76107384576194159622447676391, −6.96137540118603283713759360083, −5.89960260303806982026225720337, −4.80444389029006834066846111277, −4.05364642066810197625864499393, −3.24943994612016545147605402034, −2.26343276940070835628274811903, −1.03229455790525522683899251337,
2.34415409650363142519678841769, 3.47824823381794499381507015505, 3.84364056192509123155752047934, 4.87398953401884830423088259775, 5.73793500019226342230825119313, 7.08659214781533850246538286538, 7.36574307003351365572472196581, 8.143249841491063318164072749192, 8.954730742083777416021422936907, 9.571574671542455516364435061650