| L(s) = 1 | − 2-s − i·3-s + 4-s + i·6-s + 2i·7-s − 8-s + 2·9-s − i·12-s + 13-s − 2i·14-s + 16-s + (−4 + i)17-s − 2·18-s + 5·19-s + 2·21-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577i·3-s + 0.5·4-s + 0.408i·6-s + 0.755i·7-s − 0.353·8-s + 0.666·9-s − 0.288i·12-s + 0.277·13-s − 0.534i·14-s + 0.250·16-s + (−0.970 + 0.242i)17-s − 0.471·18-s + 1.14·19-s + 0.436·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.16392 + 0.143285i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.16392 + 0.143285i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 17 | \( 1 + (4 - i)T \) |
| good | 3 | \( 1 + iT - 3T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 9iT - 29T^{2} \) |
| 31 | \( 1 - 5iT - 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 10iT - 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 - 7T + 47T^{2} \) |
| 53 | \( 1 - T + 53T^{2} \) |
| 59 | \( 1 - 5T + 59T^{2} \) |
| 61 | \( 1 - 5iT - 61T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 - 5iT - 71T^{2} \) |
| 73 | \( 1 + 11iT - 73T^{2} \) |
| 79 | \( 1 + 16iT - 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + 5T + 89T^{2} \) |
| 97 | \( 1 - 7iT - 97T^{2} \) |
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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
−10.21130325097609687991670977440, −9.107553887859575550561392342461, −8.756521048540591506971298043076, −7.53035784887409850165878548672, −7.05816425667961143353263841968, −6.04653794066634765448798057552, −5.08304813103317134622706501522, −3.61280060199149486537874682838, −2.31295352773437916181808572249, −1.23523536068747260977172199612,
0.863703660807445950954519393600, 2.44504625383375723338089533864, 3.86941983555461124360414389346, 4.58164907461346157125032958002, 5.90910364629537649675911337615, 6.93416018122710786802240135420, 7.61092106659351958290187429860, 8.546231389257606828142390211399, 9.621571254248254110116793864211, 9.885854717852620640885876843613
