| L(s) = 1 | + i·3-s + (−2 + i)5-s + i·7-s − 9-s + 4·11-s − 6i·13-s + (−1 − 2i)15-s − 2i·17-s + 6·19-s − 21-s − 2i·23-s + (3 − 4i)25-s − i·27-s − 6·29-s + 2·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (−0.894 + 0.447i)5-s + 0.377i·7-s − 0.333·9-s + 1.20·11-s − 1.66i·13-s + (−0.258 − 0.516i)15-s − 0.485i·17-s + 1.37·19-s − 0.218·21-s − 0.417i·23-s + (0.600 − 0.800i)25-s − 0.192i·27-s − 1.11·29-s + 0.359·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.524431131\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.524431131\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (2 - i)T \) |
| 7 | \( 1 - iT \) |
| good | 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 + 2iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 4iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 14T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 16iT - 83T^{2} \) |
| 89 | \( 1 + 8T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−9.488183623370718288035170545107, −8.616106659845160174059244919632, −7.83696356226028951202082917396, −7.15798338761632556354860069909, −6.09743681843731010569910737818, −5.30397512914412908283084271049, −4.31160241874633658516906267280, −3.42031777145297627005271159277, −2.77274814962515678305357825915, −0.841417224320144161065720466590,
0.935188665223478661022326246369, 1.94256030229403833502821782546, 3.62796829751011359702272528238, 4.04747557179696972558413138932, 5.15449796315116488801900665728, 6.23263812359207440330252395833, 7.12963352568719928396862848136, 7.47284940877725177831588395027, 8.578180954717967574982679366391, 9.137980944463362249593741490558
