| L(s) = 1 | + i·3-s + (2 + i)5-s + 2i·7-s − 9-s + 6·11-s − 2i·13-s + (−1 + 2i)15-s − 6i·17-s + 4·19-s − 2·21-s + 8i·23-s + (3 + 4i)25-s − i·27-s − 8·31-s + 6i·33-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (0.894 + 0.447i)5-s + 0.755i·7-s − 0.333·9-s + 1.80·11-s − 0.554i·13-s + (−0.258 + 0.516i)15-s − 1.45i·17-s + 0.917·19-s − 0.436·21-s + 1.66i·23-s + (0.600 + 0.800i)25-s − 0.192i·27-s − 1.43·31-s + 1.04i·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
| \(L(1)\) |
\(\approx\) |
\(1.72942 + 1.06884i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.72942 + 1.06884i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 6T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 + 12iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 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
−9.891090246945847408836289396824, −9.322197807557015415522437040851, −9.012313287191673627759777057095, −7.53468707191879779860011800857, −6.71146342293275352856367529644, −5.67539042798318409561660598858, −5.19722763649408056950288916249, −3.73110619456209469904519566295, −2.89485566438997365157697238715, −1.52916421218440876802963154580,
1.12940573186945381686729139236, 1.95038766293079945737902604749, 3.62191532841093156722340039772, 4.49614591677651830024462578888, 5.76595535155158674941118526765, 6.54319991819423379909336753817, 7.09124254684315809954009350636, 8.399025752931117581040645304471, 8.973088784721920661417098146153, 9.823698858775263340021678603640
