| L(s) = 1 | − i·3-s + (−2 + i)5-s + 2i·7-s − 9-s − 2·11-s − 2i·13-s + (1 + 2i)15-s − 6i·17-s + 8·19-s + 2·21-s − 4i·23-s + (3 − 4i)25-s + i·27-s + 8·29-s + 2i·33-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + (−0.894 + 0.447i)5-s + 0.755i·7-s − 0.333·9-s − 0.603·11-s − 0.554i·13-s + (0.258 + 0.516i)15-s − 1.45i·17-s + 1.83·19-s + 0.436·21-s − 0.834i·23-s + (0.600 − 0.800i)25-s + 0.192i·27-s + 1.48·29-s + 0.348i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.991107 - 0.612537i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.991107 - 0.612537i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (2 - i)T \) |
| good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 - 8T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 12iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 10iT - 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 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
−9.856996645262923943845465558976, −8.948993885199513722150729850187, −7.971916878965730007842310926661, −7.50205887892589392545824838719, −6.61088460128492096065687493111, −5.52856469532631125930508084951, −4.70906127391465032730567318910, −3.14521360210025973005104356142, −2.61900596778946710568016469852, −0.64760855672245040754474559886,
1.18473864608628403330133231987, 3.11533291899237363460400428290, 3.95849213705274032262490020855, 4.74778390693841042544404383001, 5.67754109398773039329041281151, 6.95701144842720454553190749396, 7.75865510406062579704489499542, 8.446189733417896080430607797997, 9.372099576621777591253133841292, 10.26863020463102016845061651065
