| L(s) = 1 | + (0.913 + 1.07i)2-s + (−0.332 + 1.97i)4-s + 0.592i·5-s + 3.54·7-s + (−2.43 + 1.44i)8-s + (−0.639 + 0.540i)10-s + 2.11i·11-s + i·13-s + (3.24 + 3.83i)14-s + (−3.77 − 1.31i)16-s − 1.91·17-s + 2.26i·19-s + (−1.16 − 0.196i)20-s + (−2.28 + 1.93i)22-s + 1.43·23-s + ⋯ |
| L(s) = 1 | + (0.645 + 0.763i)2-s + (−0.166 + 0.986i)4-s + 0.264i·5-s + 1.34·7-s + (−0.860 + 0.509i)8-s + (−0.202 + 0.171i)10-s + 0.637i·11-s + 0.277i·13-s + (0.866 + 1.02i)14-s + (−0.944 − 0.327i)16-s − 0.463·17-s + 0.518i·19-s + (−0.261 − 0.0440i)20-s + (−0.486 + 0.411i)22-s + 0.299·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.509 - 0.860i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.509 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.15899 + 2.03371i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.15899 + 2.03371i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.913 - 1.07i)T \) |
| 3 | \( 1 \) |
| 13 | \( 1 - iT \) |
| good | 5 | \( 1 - 0.592iT - 5T^{2} \) |
| 7 | \( 1 - 3.54T + 7T^{2} \) |
| 11 | \( 1 - 2.11iT - 11T^{2} \) |
| 17 | \( 1 + 1.91T + 17T^{2} \) |
| 19 | \( 1 - 2.26iT - 19T^{2} \) |
| 23 | \( 1 - 1.43T + 23T^{2} \) |
| 29 | \( 1 - 0.214iT - 29T^{2} \) |
| 31 | \( 1 - 1.97T + 31T^{2} \) |
| 37 | \( 1 - 3.57iT - 37T^{2} \) |
| 41 | \( 1 + 0.377T + 41T^{2} \) |
| 43 | \( 1 - 0.319iT - 43T^{2} \) |
| 47 | \( 1 + 9.40T + 47T^{2} \) |
| 53 | \( 1 - 9.40iT - 53T^{2} \) |
| 59 | \( 1 - 2.37iT - 59T^{2} \) |
| 61 | \( 1 + 10.7iT - 61T^{2} \) |
| 67 | \( 1 + 6.49iT - 67T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 + 3.57T + 73T^{2} \) |
| 79 | \( 1 - 4.55T + 79T^{2} \) |
| 83 | \( 1 + 11.3iT - 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
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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
−10.46196807864283646313953610135, −9.253993374515846193128291547960, −8.417354486686078604856553476750, −7.71566034182092227269095821017, −6.93872645310366537645127217259, −6.06276195591203368825307876308, −4.87720022104636506754615163297, −4.51953327194965736852077727168, −3.17919875560143413424684983048, −1.85976020936456851605892587544,
0.955845712134721712839739561880, 2.17392364516266091928964770757, 3.34198509846568206202898570743, 4.56613911394159424133150966468, 5.07133885674250304033182551096, 6.05341677188260840207374008865, 7.16053159506704180324928042020, 8.383018532446776363516403966720, 8.897673583661552844522652966056, 10.01138800363008111524436882597
