| L(s) = 1 | − i·2-s − i·3-s − 4-s − 6-s − 2i·7-s + i·8-s − 9-s + 4·11-s + i·12-s − 2·14-s + 16-s + i·17-s + i·18-s − 4·19-s − 2·21-s − 4i·22-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.408·6-s − 0.755i·7-s + 0.353i·8-s − 0.333·9-s + 1.20·11-s + 0.288i·12-s − 0.534·14-s + 0.250·16-s + 0.242i·17-s + 0.235i·18-s − 0.917·19-s − 0.436·21-s − 0.852i·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.027689012\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.027689012\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 17 | \( 1 - iT \) |
| good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 + 2iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 + 14iT - 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 + 10T + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 16iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−8.669883095721105648976007691226, −7.66106952096833165147610140496, −6.97847923159231804340582986809, −6.22569187672602091647602151710, −5.29365748304624770580429840224, −4.03894930364135756807437421118, −3.77743546023904913097784416572, −2.35826950617406455265896024736, −1.52976664290050473153125459204, −0.34236392066307127855807273105,
1.56828725657156285739851042991, 2.93054097669083366032716287060, 3.96180310079848574260274650114, 4.57577009366423605685451014747, 5.71744137030866999183655710444, 6.00836077780445760355601057727, 7.05649992689755557857076166706, 7.77166329164970050362670181309, 8.803550117426026840946027100020, 9.164211803023117544138221586685
