| L(s) = 1 | + i·2-s + i·3-s − 4-s + 2.23i·5-s − 6-s + 2i·7-s − i·8-s − 9-s − 2.23·10-s + 5.23·11-s − i·12-s + 4.47i·13-s − 2·14-s − 2.23·15-s + 16-s + 4.47i·17-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.999i·5-s − 0.408·6-s + 0.755i·7-s − 0.353i·8-s − 0.333·9-s − 0.707·10-s + 1.57·11-s − 0.288i·12-s + 1.24i·13-s − 0.534·14-s − 0.577·15-s + 0.250·16-s + 1.08i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.566691179\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.566691179\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
| 5 | \( 1 - 2.23iT \) |
| 37 | \( 1 - iT \) |
| good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - 5.23T + 11T^{2} \) |
| 13 | \( 1 - 4.47iT - 13T^{2} \) |
| 17 | \( 1 - 4.47iT - 17T^{2} \) |
| 19 | \( 1 - 7.23T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 + 2.47T + 31T^{2} \) |
| 41 | \( 1 + 4.47T + 41T^{2} \) |
| 43 | \( 1 + 10.4iT - 43T^{2} \) |
| 47 | \( 1 - 9.70iT - 47T^{2} \) |
| 53 | \( 1 + 12.4iT - 53T^{2} \) |
| 59 | \( 1 + 6.94T + 59T^{2} \) |
| 61 | \( 1 - 5.23T + 61T^{2} \) |
| 67 | \( 1 + 10.4iT - 67T^{2} \) |
| 71 | \( 1 + 2.47T + 71T^{2} \) |
| 73 | \( 1 + 10.4iT - 73T^{2} \) |
| 79 | \( 1 - 2.47T + 79T^{2} \) |
| 83 | \( 1 - 15.4iT - 83T^{2} \) |
| 89 | \( 1 - 7.52T + 89T^{2} \) |
| 97 | \( 1 + 8.18iT - 97T^{2} \) |
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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
−10.02492011057624879380367825723, −9.275244858119433600004795092222, −8.825227200721764963961086411746, −7.67830117841755347187842962042, −6.70636719900031347112568444633, −6.24021152893788896918988029078, −5.26430223891565387563656588570, −4.05903672584067963898766658599, −3.42314656851286579869555511541, −1.88940493153394150275301177940,
0.78334332832706153468519125823, 1.45657001153063217589613621064, 3.11770325065176283827982959646, 3.96910525310792983259091587082, 5.08589184260348488999640549100, 5.82620419311320945768086043482, 7.20950372246305573815087607833, 7.71428384071908031495650511529, 8.830012348720911360909356834759, 9.438557589058204321290624973862
