L(s) = 1 | + 1.40i·3-s + (0.466 − 2.18i)5-s + 1.57i·7-s + 1.03·9-s − 4.35·11-s + 0.964i·13-s + (3.06 + 0.655i)15-s + 0.300i·17-s − 8.62·19-s − 2.20·21-s + i·23-s + (−4.56 − 2.04i)25-s + 5.65i·27-s − 4.76·29-s + 5.59·31-s + ⋯ |
L(s) = 1 | + 0.810i·3-s + (0.208 − 0.977i)5-s + 0.593i·7-s + 0.343·9-s − 1.31·11-s + 0.267i·13-s + (0.792 + 0.169i)15-s + 0.0728i·17-s − 1.97·19-s − 0.481·21-s + 0.208i·23-s + (−0.912 − 0.408i)25-s + 1.08i·27-s − 0.885·29-s + 1.00·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.208i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.977 - 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5587687453\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5587687453\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.466 + 2.18i)T \) |
| 23 | \( 1 - iT \) |
good | 3 | \( 1 - 1.40iT - 3T^{2} \) |
| 7 | \( 1 - 1.57iT - 7T^{2} \) |
| 11 | \( 1 + 4.35T + 11T^{2} \) |
| 13 | \( 1 - 0.964iT - 13T^{2} \) |
| 17 | \( 1 - 0.300iT - 17T^{2} \) |
| 19 | \( 1 + 8.62T + 19T^{2} \) |
| 29 | \( 1 + 4.76T + 29T^{2} \) |
| 31 | \( 1 - 5.59T + 31T^{2} \) |
| 37 | \( 1 - 4.38iT - 37T^{2} \) |
| 41 | \( 1 + 6.62T + 41T^{2} \) |
| 43 | \( 1 - 1.72iT - 43T^{2} \) |
| 47 | \( 1 + 0.687iT - 47T^{2} \) |
| 53 | \( 1 - 8.05iT - 53T^{2} \) |
| 59 | \( 1 - 5.74T + 59T^{2} \) |
| 61 | \( 1 + 13.6T + 61T^{2} \) |
| 67 | \( 1 - 6.49iT - 67T^{2} \) |
| 71 | \( 1 + 9.89T + 71T^{2} \) |
| 73 | \( 1 - 6.35iT - 73T^{2} \) |
| 79 | \( 1 + 6.95T + 79T^{2} \) |
| 83 | \( 1 - 0.185iT - 83T^{2} \) |
| 89 | \( 1 - 1.64T + 89T^{2} \) |
| 97 | \( 1 - 9.21iT - 97T^{2} \) |
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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
−9.657909994004062836382948108163, −8.759558769220624648495719436673, −8.381931220234610135478028353464, −7.36903096556763312901234452310, −6.20246470580036945684071430532, −5.43323348925000703900778943035, −4.66462719025794611322213590558, −4.10913536262885182261315263736, −2.74417565940884840030554966724, −1.68726517466984282971835180240,
0.18855380373389922085233442517, 1.87801128986824782543354927746, 2.61078090241907958418957985112, 3.76965302208144730447865095262, 4.76884876497766253405870418742, 5.95344062082535109765315940641, 6.58424217373996436782185958132, 7.34324534735109004782103335077, 7.83265654566042083130891281309, 8.690002893015110389408160472949