| L(s) = 1 | − 1.41·2-s + 2.00·4-s + (4.21 + 2.69i)5-s + 7.64i·7-s − 2.82·8-s + (−5.96 − 3.80i)10-s + 11.3i·11-s + 14.3i·13-s − 10.8i·14-s + 4.00·16-s + 17.3·17-s − 31.8·19-s + (8.42 + 5.38i)20-s − 16.0i·22-s − 16.5·23-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.500·4-s + (0.842 + 0.538i)5-s + 1.09i·7-s − 0.353·8-s + (−0.596 − 0.380i)10-s + 1.03i·11-s + 1.10i·13-s − 0.772i·14-s + 0.250·16-s + 1.01·17-s − 1.67·19-s + (0.421 + 0.269i)20-s − 0.731i·22-s − 0.718·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.842 - 0.538i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.842 - 0.538i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.080767535\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.080767535\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-4.21 - 2.69i)T \) |
| good | 7 | \( 1 - 7.64iT - 49T^{2} \) |
| 11 | \( 1 - 11.3iT - 121T^{2} \) |
| 13 | \( 1 - 14.3iT - 169T^{2} \) |
| 17 | \( 1 - 17.3T + 289T^{2} \) |
| 19 | \( 1 + 31.8T + 361T^{2} \) |
| 23 | \( 1 + 16.5T + 529T^{2} \) |
| 29 | \( 1 + 8.49iT - 841T^{2} \) |
| 31 | \( 1 + 45.8T + 961T^{2} \) |
| 37 | \( 1 + 31.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 68.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 31.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 46.0T + 2.20e3T^{2} \) |
| 53 | \( 1 - 53.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + 64.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 96.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + 1.72iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 26.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 51.4iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 59.2T + 6.24e3T^{2} \) |
| 83 | \( 1 + 6.92T + 6.88e3T^{2} \) |
| 89 | \( 1 - 115. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 158. iT - 9.40e3T^{2} \) |
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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
−10.35454450315750350717291904508, −9.248954894436011665900416147106, −9.157404345501240906496876454698, −7.86536518620339215473963660011, −6.94136182386001164398841425297, −6.15853986692548996446363916880, −5.35476307978443231162103277573, −3.91757800652630079244672924426, −2.29616049652839848334945309248, −1.92781559016223131800540414455,
0.44438765300442162992423628661, 1.49494292429696588752871626122, 2.95587327016473727387659466410, 4.15158387633864804878400655955, 5.52854906370031555604729446599, 6.14670847263423014689556407995, 7.27103215781473321790822935211, 8.183915358461418397852511321960, 8.759897129222548661428952567916, 9.845635631003995265172976802396
