L(s) = 1 | + (101. + 78.2i)2-s + 3.47e3·3-s + (4.14e3 + 1.58e4i)4-s + 7.83e4i·5-s + (3.52e5 + 2.71e5i)6-s − 1.22e6i·7-s + (−8.19e5 + 1.93e6i)8-s + 7.30e6·9-s + (−6.12e6 + 7.93e6i)10-s − 1.24e7·11-s + (1.44e7 + 5.50e7i)12-s + 6.40e6i·13-s + (9.56e7 − 1.23e8i)14-s + 2.72e8i·15-s + (−2.34e8 + 1.31e8i)16-s + 5.71e8·17-s + ⋯ |
L(s) = 1 | + (0.791 + 0.611i)2-s + 1.58·3-s + (0.253 + 0.967i)4-s + 1.00i·5-s + (1.25 + 0.971i)6-s − 1.48i·7-s + (−0.390 + 0.920i)8-s + 1.52·9-s + (−0.612 + 0.793i)10-s − 0.637·11-s + (0.402 + 1.53i)12-s + 0.102i·13-s + (0.907 − 1.17i)14-s + 1.59i·15-s + (−0.871 + 0.489i)16-s + 1.39·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.390 - 0.920i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (0.390 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{15}{2})\) |
\(\approx\) |
\(3.40369 + 2.25278i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.40369 + 2.25278i\) |
\(L(8)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-101. - 78.2i)T \) |
good | 3 | \( 1 - 3.47e3T + 4.78e6T^{2} \) |
| 5 | \( 1 - 7.83e4iT - 6.10e9T^{2} \) |
| 7 | \( 1 + 1.22e6iT - 6.78e11T^{2} \) |
| 11 | \( 1 + 1.24e7T + 3.79e14T^{2} \) |
| 13 | \( 1 - 6.40e6iT - 3.93e15T^{2} \) |
| 17 | \( 1 - 5.71e8T + 1.68e17T^{2} \) |
| 19 | \( 1 + 8.25e8T + 7.99e17T^{2} \) |
| 23 | \( 1 + 5.33e9iT - 1.15e19T^{2} \) |
| 29 | \( 1 - 1.07e10iT - 2.97e20T^{2} \) |
| 31 | \( 1 + 3.45e10iT - 7.56e20T^{2} \) |
| 37 | \( 1 + 2.16e10iT - 9.01e21T^{2} \) |
| 41 | \( 1 - 1.16e10T + 3.79e22T^{2} \) |
| 43 | \( 1 + 3.57e11T + 7.38e22T^{2} \) |
| 47 | \( 1 - 8.17e10iT - 2.56e23T^{2} \) |
| 53 | \( 1 - 7.30e11iT - 1.37e24T^{2} \) |
| 59 | \( 1 - 2.62e12T + 6.19e24T^{2} \) |
| 61 | \( 1 - 4.54e12iT - 9.87e24T^{2} \) |
| 67 | \( 1 + 1.32e12T + 3.67e25T^{2} \) |
| 71 | \( 1 - 8.09e12iT - 8.27e25T^{2} \) |
| 73 | \( 1 - 8.89e12T + 1.22e26T^{2} \) |
| 79 | \( 1 - 1.21e13iT - 3.68e26T^{2} \) |
| 83 | \( 1 + 2.46e13T + 7.36e26T^{2} \) |
| 89 | \( 1 - 2.19e13T + 1.95e27T^{2} \) |
| 97 | \( 1 + 1.90e13T + 6.52e27T^{2} \) |
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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
−18.68813108300011078141590068491, −16.64365783493483403588710018458, −14.86042051125701861517604910402, −14.24491757057492284851632660357, −13.10141726470029430547089779411, −10.43316916343990394877301909888, −8.083593495805472665591564292384, −6.99233649864700132965454365112, −3.97327993666472933887283399769, −2.72635363297924544795326763198,
1.80947212579553625518869336004, 3.20984289227199090668779851926, 5.26950396484665864937630375594, 8.375909457768375487348363375263, 9.622227863294866990670434664311, 12.22027918934675384703203605895, 13.27848617547516535311391120506, 14.75244285927271918079983358798, 15.74002535080821699883064888254, 18.71154765235210042296813589791