L(s) = 1 | + (1.28 − 2.21i)3-s − 3.56·5-s + (−1.28 − 2.21i)7-s + (−1.78 − 3.08i)9-s + (1.28 − 2.21i)11-s + (3.34 − 1.35i)13-s + (−4.56 + 7.90i)15-s + (2.5 + 4.33i)17-s + (−1.28 − 2.21i)19-s − 6.56·21-s + (−1.84 + 3.19i)23-s + 7.68·25-s − 1.43·27-s + (2.5 − 4.33i)29-s + 8·31-s + ⋯ |
L(s) = 1 | + (0.739 − 1.28i)3-s − 1.59·5-s + (−0.484 − 0.838i)7-s + (−0.593 − 1.02i)9-s + (0.386 − 0.668i)11-s + (0.926 − 0.375i)13-s + (−1.17 + 2.03i)15-s + (0.606 + 1.05i)17-s + (−0.293 − 0.508i)19-s − 1.43·21-s + (−0.384 + 0.665i)23-s + 1.53·25-s − 0.276·27-s + (0.464 − 0.804i)29-s + 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.396 + 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.396 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.609743 - 0.927161i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.609743 - 0.927161i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 13 | \( 1 + (-3.34 + 1.35i)T \) |
good | 3 | \( 1 + (-1.28 + 2.21i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + 3.56T + 5T^{2} \) |
| 7 | \( 1 + (1.28 + 2.21i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.28 + 2.21i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-2.5 - 4.33i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.28 + 2.21i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.84 - 3.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.5 + 4.33i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (4.62 - 8.00i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.28 + 5.68i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 - 4.43T + 53T^{2} \) |
| 59 | \( 1 + (1.28 + 2.21i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.62 - 6.27i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4.71 - 8.17i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.84 - 6.65i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 1.31T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 2.24T + 83T^{2} \) |
| 89 | \( 1 + (-4.84 + 8.38i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.40 + 2.43i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−12.12949948605614573381435372968, −11.40991734360945766310133936896, −10.22835666034072029489066757988, −8.434946437464978969908515701753, −8.175736939851232805285022758318, −7.16133893650261867947405429306, −6.27696859844700661123572410714, −4.03028902845766978729060621019, −3.20630090067955042991052838367, −0.965947150351899980744876343975,
3.02291831456493942594435327088, 3.92526167837879415169695317285, 4.85364899103391681345091609226, 6.63038039199585710554936059236, 8.046297083190571254615315119041, 8.759783905969369412105160659711, 9.637582294713974755028482191050, 10.65234987295097357779089022588, 11.82640739308588557410491844190, 12.33002875267007939038628731433