| L(s) = 1 | − i·2-s + i·3-s − 4-s + 6-s + i·8-s − 9-s + 6·11-s − i·12-s − 2i·13-s + 16-s − i·17-s + i·18-s − 4·19-s − 6i·22-s − 5i·23-s − 24-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s + 0.353i·8-s − 0.333·9-s + 1.80·11-s − 0.288i·12-s − 0.554i·13-s + 0.250·16-s − 0.242i·17-s + 0.235i·18-s − 0.917·19-s − 1.27i·22-s − 1.04i·23-s − 0.204·24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.726307164\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.726307164\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 6T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 5iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 3iT - 37T^{2} \) |
| 41 | \( 1 - 5T + 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - iT - 53T^{2} \) |
| 59 | \( 1 - 3T + 59T^{2} \) |
| 61 | \( 1 - 5T + 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 - 5T + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + iT - 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 + 16iT - 97T^{2} \) |
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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
−8.861227858636236296327715831911, −8.404287303391963351279336309429, −7.21132089484149374748748345031, −6.35024803787895492736169678760, −5.56892643260902236513624021552, −4.42828915703440611846751360443, −4.02576201808408322378946990158, −3.05863375921527275644362115194, −1.99834101287055966919933451227, −0.69845961306140411679436682715,
1.07733731989298990584220617126, 2.10301667276574041648777851317, 3.62986185744889399820409105794, 4.19798728007053181961321240864, 5.31373220415078402401210557398, 6.24335644334790230713211404676, 6.65418256735679247962642165776, 7.37579653310079361538377602996, 8.210294379297691321538279326149, 9.029613404362900978178823490171
