| L(s) = 1 | + (19.5 + 11.3i)2-s + (255. + 443. i)4-s + (1.76e3 − 1.02e3i)5-s + (−1.03e4 + 1.78e4i)7-s + 1.15e4i·8-s + 4.61e4·10-s + (2.99e4 + 1.73e4i)11-s + (−2.15e5 − 3.73e5i)13-s + (−4.04e5 + 2.33e5i)14-s + (−1.31e5 + 2.27e5i)16-s − 2.41e6i·17-s + 3.75e6·19-s + (9.04e5 + 5.22e5i)20-s + (3.91e5 + 6.78e5i)22-s + (6.81e6 − 3.93e6i)23-s + ⋯ |
| L(s) = 1 | + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (0.565 − 0.326i)5-s + (−0.613 + 1.06i)7-s + 0.353i·8-s + 0.461·10-s + (0.186 + 0.107i)11-s + (−0.581 − 1.00i)13-s + (−0.751 + 0.434i)14-s + (−0.125 + 0.216i)16-s − 1.69i·17-s + 1.51·19-s + (0.282 + 0.163i)20-s + (0.0760 + 0.131i)22-s + (1.05 − 0.611i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(3.447284108\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.447284108\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-19.5 - 11.3i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-1.76e3 + 1.02e3i)T + (4.88e6 - 8.45e6i)T^{2} \) |
| 7 | \( 1 + (1.03e4 - 1.78e4i)T + (-1.41e8 - 2.44e8i)T^{2} \) |
| 11 | \( 1 + (-2.99e4 - 1.73e4i)T + (1.29e10 + 2.24e10i)T^{2} \) |
| 13 | \( 1 + (2.15e5 + 3.73e5i)T + (-6.89e10 + 1.19e11i)T^{2} \) |
| 17 | \( 1 + 2.41e6iT - 2.01e12T^{2} \) |
| 19 | \( 1 - 3.75e6T + 6.13e12T^{2} \) |
| 23 | \( 1 + (-6.81e6 + 3.93e6i)T + (2.07e13 - 3.58e13i)T^{2} \) |
| 29 | \( 1 + (2.09e7 + 1.21e7i)T + (2.10e14 + 3.64e14i)T^{2} \) |
| 31 | \( 1 + (-1.79e7 - 3.11e7i)T + (-4.09e14 + 7.09e14i)T^{2} \) |
| 37 | \( 1 - 2.89e7T + 4.80e15T^{2} \) |
| 41 | \( 1 + (-8.87e7 + 5.12e7i)T + (6.71e15 - 1.16e16i)T^{2} \) |
| 43 | \( 1 + (8.64e7 - 1.49e8i)T + (-1.08e16 - 1.87e16i)T^{2} \) |
| 47 | \( 1 + (-1.45e8 - 8.37e7i)T + (2.62e16 + 4.55e16i)T^{2} \) |
| 53 | \( 1 + 4.72e8iT - 1.74e17T^{2} \) |
| 59 | \( 1 + (1.24e8 - 7.16e7i)T + (2.55e17 - 4.42e17i)T^{2} \) |
| 61 | \( 1 + (-6.50e8 + 1.12e9i)T + (-3.56e17 - 6.17e17i)T^{2} \) |
| 67 | \( 1 + (-9.08e8 - 1.57e9i)T + (-9.11e17 + 1.57e18i)T^{2} \) |
| 71 | \( 1 + 2.13e8iT - 3.25e18T^{2} \) |
| 73 | \( 1 - 1.94e9T + 4.29e18T^{2} \) |
| 79 | \( 1 + (-1.14e9 + 1.98e9i)T + (-4.73e18 - 8.19e18i)T^{2} \) |
| 83 | \( 1 + (-5.22e9 - 3.01e9i)T + (7.75e18 + 1.34e19i)T^{2} \) |
| 89 | \( 1 - 6.12e9iT - 3.11e19T^{2} \) |
| 97 | \( 1 + (1.69e9 - 2.94e9i)T + (-3.68e19 - 6.38e19i)T^{2} \) |
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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
−11.33048227787597273107779337305, −9.702242490770567220580061174212, −9.210162742964120167421609401981, −7.78455173330961181688212650089, −6.73041602573908630379164003856, −5.47057665853589842510253005190, −5.07040624544423460086467417707, −3.20730976029278608947359524453, −2.47195202190919084781631724671, −0.75764974872371545629448149378,
0.889044618273897901403328361763, 2.03196516666276159284658787456, 3.37815895943560798307129739136, 4.23560599111415363007984170275, 5.64371286863316353013742295429, 6.63533407313489780606677344196, 7.52324229449210048233498334894, 9.289020586333302127315474640934, 10.03742974817951806589683620617, 10.92359210589522749361222212089
