| L(s) = 1 | + 2.17i·3-s + (1.45 − 1.69i)5-s − i·7-s − 1.74·9-s − 2.13·11-s + 5.52i·13-s + (3.68 + 3.17i)15-s + 5.24i·17-s − 6.50·19-s + 2.17·21-s − 6.35i·23-s + (−0.741 − 4.94i)25-s + 2.74i·27-s + 9.09·29-s − 6.77·31-s + ⋯ |
| L(s) = 1 | + 1.25i·3-s + (0.652 − 0.757i)5-s − 0.377i·7-s − 0.580·9-s − 0.643·11-s + 1.53i·13-s + (0.952 + 0.820i)15-s + 1.27i·17-s − 1.49·19-s + 0.475·21-s − 1.32i·23-s + (−0.148 − 0.988i)25-s + 0.527i·27-s + 1.68·29-s − 1.21·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.757 - 0.652i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.757 - 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.355894321\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.355894321\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.45 + 1.69i)T \) |
| 7 | \( 1 + iT \) |
| good | 3 | \( 1 - 2.17iT - 3T^{2} \) |
| 11 | \( 1 + 2.13T + 11T^{2} \) |
| 13 | \( 1 - 5.52iT - 13T^{2} \) |
| 17 | \( 1 - 5.24iT - 17T^{2} \) |
| 19 | \( 1 + 6.50T + 19T^{2} \) |
| 23 | \( 1 + 6.35iT - 23T^{2} \) |
| 29 | \( 1 - 9.09T + 29T^{2} \) |
| 31 | \( 1 + 6.77T + 31T^{2} \) |
| 37 | \( 1 - 9.88iT - 37T^{2} \) |
| 41 | \( 1 + 2.35T + 41T^{2} \) |
| 43 | \( 1 - 0.354iT - 43T^{2} \) |
| 47 | \( 1 - 8.57iT - 47T^{2} \) |
| 53 | \( 1 - 7.37iT - 53T^{2} \) |
| 59 | \( 1 - 6.50T + 59T^{2} \) |
| 61 | \( 1 + 5.43T + 61T^{2} \) |
| 67 | \( 1 - 5.48iT - 67T^{2} \) |
| 71 | \( 1 - 3.11T + 71T^{2} \) |
| 73 | \( 1 - 7.37iT - 73T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 - 11.1iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 0.979iT - 97T^{2} \) |
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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
−9.359630118860842092787269486850, −8.651229488371713988413874420020, −8.217745393084604933479677710216, −6.72962440815421420891555541626, −6.21816154639494592687210656453, −5.09613198786315794877881718796, −4.39154084841189800424365391008, −4.08052208905071201979846910398, −2.60631874975498016254656693364, −1.50744115460158168926659031559,
0.44368318902205629116568611012, 1.92866225287688273440355583566, 2.56298615490994706120610519344, 3.45583631472964650069996449569, 5.10027675750336731469683203962, 5.69587126122994161844855507703, 6.49842474328999795425319955418, 7.21001201544412380920997362750, 7.77006987696746887406464531825, 8.554812149854008626947378546615
