| L(s) = 1 | + i·2-s − 4-s + (0.224 − 2.22i)5-s − i·7-s − i·8-s + (2.22 + 0.224i)10-s − 4.89·11-s + 0.449i·13-s + 14-s + 16-s − 2i·17-s − 6.44·19-s + (−0.224 + 2.22i)20-s − 4.89i·22-s − 6.89i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (0.100 − 0.994i)5-s − 0.377i·7-s − 0.353i·8-s + (0.703 + 0.0710i)10-s − 1.47·11-s + 0.124i·13-s + 0.267·14-s + 0.250·16-s − 0.485i·17-s − 1.47·19-s + (−0.0502 + 0.497i)20-s − 1.04i·22-s − 1.43i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.100 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.486110 - 0.537691i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.486110 - 0.537691i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.224 + 2.22i)T \) |
| 7 | \( 1 + iT \) |
| good | 11 | \( 1 + 4.89T + 11T^{2} \) |
| 13 | \( 1 - 0.449iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 6.44T + 19T^{2} \) |
| 23 | \( 1 + 6.89iT - 23T^{2} \) |
| 29 | \( 1 + 2.89T + 29T^{2} \) |
| 31 | \( 1 + 0.898T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 + 8.89iT - 43T^{2} \) |
| 47 | \( 1 - 0.898iT - 47T^{2} \) |
| 53 | \( 1 + 1.10iT - 53T^{2} \) |
| 59 | \( 1 + 6.44T + 59T^{2} \) |
| 61 | \( 1 - 8.44T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 - 6.89iT - 73T^{2} \) |
| 79 | \( 1 - 2.89T + 79T^{2} \) |
| 83 | \( 1 - 2.44iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 3.79iT - 97T^{2} \) |
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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
−10.29500671879199061593691807271, −9.315417303594220658446811115572, −8.445486585513488408837819519257, −7.86667197692029829221632365176, −6.81273101070116668849147384219, −5.74930566708355336079468198487, −4.89426595155692560178960366074, −4.10663231787645019740239995239, −2.34857903561652667518396159725, −0.37207752980453170456608730935,
2.06291773682831641353908691716, 2.91603693250900677464013527327, 4.04353895997751936555563088732, 5.37528065804666157202468237752, 6.18328769111223963362739082582, 7.47918227190992493687800604333, 8.177152183413441176061524876077, 9.335147133517114911889018881894, 10.15156783435939632554681246991, 10.90018181514141473123971025272
