L(s) = 1 | + (0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (−0.499 − 0.866i)6-s + (1.36 + 2.36i)7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (−0.366 + 0.633i)11-s − 0.999·12-s + (−0.232 − 3.59i)13-s + 2.73·14-s + (−0.5 + 0.866i)16-s + (2.23 + 3.86i)17-s − 0.999·18-s + (−0.633 − 1.09i)19-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.204 − 0.353i)6-s + (0.516 + 0.894i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.110 + 0.191i)11-s − 0.288·12-s + (−0.0643 − 0.997i)13-s + 0.730·14-s + (−0.125 + 0.216i)16-s + (0.541 + 0.937i)17-s − 0.235·18-s + (−0.145 − 0.251i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.202 + 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.202 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.346057597\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.346057597\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (0.232 + 3.59i)T \) |
good | 7 | \( 1 + (-1.36 - 2.36i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.366 - 0.633i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-2.23 - 3.86i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.633 + 1.09i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.09 + 5.36i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.23 + 5.59i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + (-3.96 + 6.86i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.59 + 7.96i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.63 + 2.83i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 7.66T + 47T^{2} \) |
| 53 | \( 1 + 7.73T + 53T^{2} \) |
| 59 | \( 1 + (-6.19 - 10.7i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.06 - 8.76i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.83 + 6.63i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.633 + 1.09i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 4.66T + 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 + 5.26T + 83T^{2} \) |
| 89 | \( 1 + (-2.46 + 4.26i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (5 + 8.66i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.820424217333315020464966976341, −8.324540576681515212155035348229, −7.56162434245775259836272953696, −6.40142870204875971390998903245, −5.70499125722044162871030313679, −4.90769610475150645729936164571, −3.87976489547270539604290728719, −2.74422042506076475094974567877, −2.16650057307795582869863523196, −0.820771662027204888197696014781,
1.28911704080273454261365840138, 2.89446939770363416851386006523, 3.72471667165049509052633006044, 4.72592463605759589454873799262, 5.06470395111084144661073976653, 6.36203185160931943859186275328, 7.03058868286751895464099871809, 7.88357785342434779271989359988, 8.393620296080588019688529311825, 9.570359221571557059542810648105