L(s) = 1 | + 1.73i·2-s + 1.73i·3-s + 1.00·4-s + 4·5-s − 2.99·6-s − 10·7-s + 8.66i·8-s − 2.99·9-s + 6.92i·10-s + 10·11-s + 1.73i·12-s − 24.2i·13-s − 17.3i·14-s + 6.92i·15-s − 10.9·16-s + 10·17-s + ⋯ |
L(s) = 1 | + 0.866i·2-s + 0.577i·3-s + 0.250·4-s + 0.800·5-s − 0.499·6-s − 1.42·7-s + 1.08i·8-s − 0.333·9-s + 0.692i·10-s + 0.909·11-s + 0.144i·12-s − 1.86i·13-s − 1.23i·14-s + 0.461i·15-s − 0.687·16-s + 0.588·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.930494 + 0.930494i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.930494 + 0.930494i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 1.73iT \) |
| 19 | \( 1 - 19T \) |
good | 2 | \( 1 - 1.73iT - 4T^{2} \) |
| 5 | \( 1 - 4T + 25T^{2} \) |
| 7 | \( 1 + 10T + 49T^{2} \) |
| 11 | \( 1 - 10T + 121T^{2} \) |
| 13 | \( 1 + 24.2iT - 169T^{2} \) |
| 17 | \( 1 - 10T + 289T^{2} \) |
| 23 | \( 1 + 20T + 529T^{2} \) |
| 29 | \( 1 + 34.6iT - 841T^{2} \) |
| 31 | \( 1 - 17.3iT - 961T^{2} \) |
| 37 | \( 1 + 10.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 34.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 10T + 1.84e3T^{2} \) |
| 47 | \( 1 + 80T + 2.20e3T^{2} \) |
| 53 | \( 1 - 41.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 34.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 10T + 3.72e3T^{2} \) |
| 67 | \( 1 + 76.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 103. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 10T + 5.32e3T^{2} \) |
| 79 | \( 1 + 17.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 70T + 6.88e3T^{2} \) |
| 89 | \( 1 + 103. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 76.2iT - 9.40e3T^{2} \) |
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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
−15.49156123562482323109887776353, −14.38955928274538277159436663122, −13.27403678147050223278945494993, −11.89901459986341513567072503947, −10.26332203093031496937718544473, −9.522587216109302787336900249024, −7.897302096499137287891559620458, −6.33572648175175447126372369560, −5.59048006745562690375790626751, −3.15472658719876178214348458808,
1.78574448882344567633074320409, 3.51234070508552345571568617431, 6.21793170496981181423693146114, 6.95830526604141885750351687772, 9.322590548795064599646673860156, 9.894234039556831273750288864233, 11.54705582306639837783315501235, 12.27637028248028792295121542288, 13.39349028415373568368380864033, 14.26780274337332871431694253018