| L(s) = 1 | − 3-s + (1 + 2i)5-s − 2i·7-s + 9-s − 2i·11-s + 2·13-s + (−1 − 2i)15-s − 4i·17-s + 2i·19-s + 2i·21-s − 6i·23-s + (−3 + 4i)25-s − 27-s + 4i·29-s + 8·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + (0.447 + 0.894i)5-s − 0.755i·7-s + 0.333·9-s − 0.603i·11-s + 0.554·13-s + (−0.258 − 0.516i)15-s − 0.970i·17-s + 0.458i·19-s + 0.436i·21-s − 1.25i·23-s + (−0.600 + 0.800i)25-s − 0.192·27-s + 0.742i·29-s + 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.41435 - 0.229517i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.41435 - 0.229517i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + (-1 - 2i)T \) |
| good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 - 4iT - 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 - 10T + 53T^{2} \) |
| 59 | \( 1 - 6iT - 59T^{2} \) |
| 61 | \( 1 + 8iT - 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 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
−10.31761260943870157738625025801, −9.311777268275246627115943639247, −8.234025792273744250414436771549, −7.24960040314863171584075486849, −6.53311196146945867908972306599, −5.85859111757882876564039662290, −4.72559186444712509004153418741, −3.64267859417867646092280427371, −2.54811782788153196829914844145, −0.884294251367325343545835127185,
1.18778929982582805162275796947, 2.38800983033393017202192271592, 4.02680326529006076127917706063, 4.89993980612448765572676202232, 5.81459289361072466628722626456, 6.32643152367091625855879949501, 7.64814639973234439515610557836, 8.481274652820164988815424914377, 9.316612297881715635610248252856, 9.944150576924295454554433021481
