| L(s) = 1 | + 2-s + 4-s − 5-s + 2.80i·7-s + 8-s − 10-s − 4.71i·11-s + 1.28·13-s + 2.80i·14-s + 16-s + 4.50i·17-s − 0.354i·19-s − 20-s − 4.71i·22-s − 7.77i·23-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.447·5-s + 1.05i·7-s + 0.353·8-s − 0.316·10-s − 1.42i·11-s + 0.355·13-s + 0.749i·14-s + 0.250·16-s + 1.09i·17-s − 0.0813i·19-s − 0.223·20-s − 1.00i·22-s − 1.62i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.163i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.876361163\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.876361163\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 43 | \( 1 + (-2.86 + 5.90i)T \) |
| good | 7 | \( 1 - 2.80iT - 7T^{2} \) |
| 11 | \( 1 + 4.71iT - 11T^{2} \) |
| 13 | \( 1 - 1.28T + 13T^{2} \) |
| 17 | \( 1 - 4.50iT - 17T^{2} \) |
| 19 | \( 1 + 0.354iT - 19T^{2} \) |
| 23 | \( 1 + 7.77iT - 23T^{2} \) |
| 29 | \( 1 - 3.89T + 29T^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 - 5.79iT - 37T^{2} \) |
| 41 | \( 1 - 3.27iT - 41T^{2} \) |
| 47 | \( 1 - 2.52iT - 47T^{2} \) |
| 53 | \( 1 + 10.8iT - 53T^{2} \) |
| 59 | \( 1 - 6.37iT - 59T^{2} \) |
| 61 | \( 1 - 12.8iT - 61T^{2} \) |
| 67 | \( 1 - 5.09T + 67T^{2} \) |
| 71 | \( 1 - 5.02T + 71T^{2} \) |
| 73 | \( 1 - 3.80iT - 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 + 9.56iT - 83T^{2} \) |
| 89 | \( 1 - 11.8T + 89T^{2} \) |
| 97 | \( 1 + 13.2T + 97T^{2} \) |
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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
−8.474843808488179182799637909127, −7.979516426993455872738220875932, −6.66081462435036999895893659082, −6.24562982404644484712871618163, −5.57422618224981510974053901439, −4.70914398342487452616300693766, −3.87438607710033733188104906339, −3.04394445067494469663021136632, −2.33622695300237053678070991412, −0.906503620994146276490778512453,
0.890307544624750885187528337906, 2.05937326009392104880538879072, 3.16973349125292721471108432997, 3.95837244652611184281398081210, 4.60085931874719606982731763770, 5.23693635756733237192452035164, 6.32461535687823665736491116564, 7.11986221866712757296447295743, 7.46075447214891596242464195719, 8.170789141888197082501587686508
