| L(s) = 1 | + i·2-s + i·3-s − 4-s − 6-s − i·7-s − i·8-s − 9-s − 5·11-s − i·12-s + i·13-s + 14-s + 16-s + 5i·17-s − i·18-s + 21-s − 5i·22-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.408·6-s − 0.377i·7-s − 0.353i·8-s − 0.333·9-s − 1.50·11-s − 0.288i·12-s + 0.277i·13-s + 0.267·14-s + 0.250·16-s + 1.21i·17-s − 0.235i·18-s + 0.218·21-s − 1.06i·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3982971107\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3982971107\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - iT \) |
| good | 7 | \( 1 + iT - 7T^{2} \) |
| 11 | \( 1 + 5T + 11T^{2} \) |
| 17 | \( 1 - 5iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 7T + 29T^{2} \) |
| 31 | \( 1 + 9T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 + 9iT - 47T^{2} \) |
| 53 | \( 1 + 11iT - 53T^{2} \) |
| 59 | \( 1 + T + 59T^{2} \) |
| 61 | \( 1 + 7T + 61T^{2} \) |
| 67 | \( 1 + 15iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 9iT - 83T^{2} \) |
| 89 | \( 1 + 16T + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−8.880145600659618493736890585642, −8.268475352722482195552890199545, −7.51944799593593933739739001118, −6.72851077647448023587886233824, −5.69019976214886025208218714122, −5.16985933931257427817887475425, −4.18747228020501973519725178121, −3.41260050546260402018528298376, −2.07935729728428045054184120617, −0.14732165815762724911452578913,
1.27164726816683742350119633728, 2.68248858013743886723859883368, 2.91802923056800729184531683581, 4.48536724697619186947534976668, 5.25164082612100078630129655200, 5.99122045962487592630226658800, 7.16495357073504651045490759837, 7.81435049446838867369636564597, 8.560268597684493583990931931554, 9.358836541444472762023956318257
