L(s) = 1 | + (−296. + 207. i)2-s + (2.15e3 + 1.11e4i)3-s + (4.51e4 − 1.23e5i)4-s − 1.59e6i·5-s + (−2.95e6 − 2.86e6i)6-s − 3.40e6i·7-s + (1.20e7 + 4.58e7i)8-s + (−1.19e8 + 4.81e7i)9-s + (3.30e8 + 4.73e8i)10-s + 4.20e8·11-s + (1.47e9 + 2.38e8i)12-s − 2.89e9·13-s + (7.05e8 + 1.01e9i)14-s + (1.77e10 − 3.44e9i)15-s + (−1.31e10 − 1.11e10i)16-s + 2.65e10i·17-s + ⋯ |
L(s) = 1 | + (−0.819 + 0.572i)2-s + (0.189 + 0.981i)3-s + (0.344 − 0.938i)4-s − 1.82i·5-s + (−0.717 − 0.696i)6-s − 0.223i·7-s + (0.254 + 0.966i)8-s + (−0.927 + 0.372i)9-s + (1.04 + 1.49i)10-s + 0.591·11-s + (0.987 + 0.160i)12-s − 0.985·13-s + (0.127 + 0.183i)14-s + (1.79 − 0.346i)15-s + (−0.762 − 0.646i)16-s + 0.924i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.160i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (-0.987 - 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(9)\) |
\(\approx\) |
\(0.0333144 + 0.413616i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0333144 + 0.413616i\) |
\(L(\frac{19}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (296. - 207. i)T \) |
| 3 | \( 1 + (-2.15e3 - 1.11e4i)T \) |
good | 5 | \( 1 + 1.59e6iT - 7.62e11T^{2} \) |
| 7 | \( 1 + 3.40e6iT - 2.32e14T^{2} \) |
| 11 | \( 1 - 4.20e8T + 5.05e17T^{2} \) |
| 13 | \( 1 + 2.89e9T + 8.65e18T^{2} \) |
| 17 | \( 1 - 2.65e10iT - 8.27e20T^{2} \) |
| 19 | \( 1 - 1.07e11iT - 5.48e21T^{2} \) |
| 23 | \( 1 - 1.73e10T + 1.41e23T^{2} \) |
| 29 | \( 1 + 2.16e12iT - 7.25e24T^{2} \) |
| 31 | \( 1 - 6.70e12iT - 2.25e25T^{2} \) |
| 37 | \( 1 + 1.76e13T + 4.56e26T^{2} \) |
| 41 | \( 1 - 1.33e13iT - 2.61e27T^{2} \) |
| 43 | \( 1 - 1.66e13iT - 5.87e27T^{2} \) |
| 47 | \( 1 + 2.06e13T + 2.66e28T^{2} \) |
| 53 | \( 1 - 4.56e14iT - 2.05e29T^{2} \) |
| 59 | \( 1 + 2.10e15T + 1.27e30T^{2} \) |
| 61 | \( 1 + 6.78e14T + 2.24e30T^{2} \) |
| 67 | \( 1 - 4.32e15iT - 1.10e31T^{2} \) |
| 71 | \( 1 - 3.83e15T + 2.96e31T^{2} \) |
| 73 | \( 1 + 3.10e15T + 4.74e31T^{2} \) |
| 79 | \( 1 - 1.62e16iT - 1.81e32T^{2} \) |
| 83 | \( 1 + 1.61e16T + 4.21e32T^{2} \) |
| 89 | \( 1 + 4.61e16iT - 1.37e33T^{2} \) |
| 97 | \( 1 + 4.10e16T + 5.95e33T^{2} \) |
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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
−16.78104449979941915751852823991, −15.60170583550950271064575349401, −14.20552536016729853119299010590, −12.15371023474209404476341615492, −10.19176640177940484112122254780, −9.102180496889161048135320462593, −8.101698344343895956170972554476, −5.62804338658008885636266597188, −4.36958143728807594143724641050, −1.45952945676638169873047283521,
0.18413387784789912276949691429, 2.19574112715782467422484380200, 3.07970635396073497755913072135, 6.71392139343936917861590235012, 7.45571850647558796747669720082, 9.357521099725031009898303485421, 11.00071009325388116339730025161, 11.99199838568310029575722390100, 13.76245451147600694514704121662, 15.11321757902218946468395435064