| L(s) = 1 | + 3-s − 5-s − 7-s + 9-s + 11-s − 13-s − 15-s + 17-s + 19-s − 21-s − 23-s + 25-s + 27-s − 29-s − 31-s + 33-s + 35-s − 37-s − 39-s + 41-s + 43-s − 45-s − 47-s + 49-s + 51-s − 53-s − 55-s + ⋯ |
| L(s) = 1 | + 3-s − 5-s − 7-s + 9-s + 11-s − 13-s − 15-s + 17-s + 19-s − 21-s − 23-s + 25-s + 27-s − 29-s − 31-s + 33-s + 35-s − 37-s − 39-s + 41-s + 43-s − 45-s − 47-s + 49-s + 51-s − 53-s − 55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.100421409\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.100421409\) |
| \(L(1)\) |
\(\approx\) |
\(1.110720734\) |
| \(L(1)\) |
\(\approx\) |
\(1.110720734\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 + T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−48.74068793215534022752522930410, −47.6209798877567549178563249871, −45.9510335880281704577596028993, −43.9346152766520649240665150339, −42.752615477159649672801727607675, −41.53981595511799883077360485981, −39.16750681241490296713715528102, −38.0617400808002403731726223796, −36.23116033995471923810665296105, −34.98256753712872018308348058124, −32.53233400657462120871686846132, −31.45201345414038079725886531059, −29.857791921026421623346922599858, −27.5181259144140394812087660182, −26.089470593190540959426835382352, −24.41964822006060936671535842043, −22.37466756829608885834718488552, −20.07363842306806925540295209504, −19.08644166006480394769355338242, −16.16469366756340984309856870967, −14.49097192816054725085740936823, −12.3405011590722115530614093056, −9.50320196197290897854598873093, −7.43447295737022101391739084615, −3.57615483678758907557757872995,
3.57615483678758907557757872995, 7.43447295737022101391739084615, 9.50320196197290897854598873093, 12.3405011590722115530614093056, 14.49097192816054725085740936823, 16.16469366756340984309856870967, 19.08644166006480394769355338242, 20.07363842306806925540295209504, 22.37466756829608885834718488552, 24.41964822006060936671535842043, 26.089470593190540959426835382352, 27.5181259144140394812087660182, 29.857791921026421623346922599858, 31.45201345414038079725886531059, 32.53233400657462120871686846132, 34.98256753712872018308348058124, 36.23116033995471923810665296105, 38.0617400808002403731726223796, 39.16750681241490296713715528102, 41.53981595511799883077360485981, 42.752615477159649672801727607675, 43.9346152766520649240665150339, 45.9510335880281704577596028993, 47.6209798877567549178563249871, 48.74068793215534022752522930410
