L(s) = 1 | − 1.41i·3-s − 2.23i·5-s + (−1.58 − 2.12i)7-s + 0.999·9-s − 3.16·15-s + (−3 + 2.23i)21-s − 9.48·23-s − 5.00·25-s − 5.65i·27-s + 6·29-s + (−4.74 + 3.53i)35-s + 4.47i·41-s + 3.16·43-s − 2.23i·45-s − 9.89i·47-s + ⋯ |
L(s) = 1 | − 0.816i·3-s − 0.999i·5-s + (−0.597 − 0.801i)7-s + 0.333·9-s − 0.816·15-s + (−0.654 + 0.487i)21-s − 1.97·23-s − 1.00·25-s − 1.08i·27-s + 1.11·29-s + (−0.801 + 0.597i)35-s + 0.698i·41-s + 0.482·43-s − 0.333i·45-s − 1.44i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.801 + 0.597i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.801 + 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.371891 - 1.12123i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.371891 - 1.12123i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 7 | \( 1 + (1.58 + 2.12i)T \) |
good | 3 | \( 1 + 1.41iT - 3T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 9.48T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 4.47iT - 41T^{2} \) |
| 43 | \( 1 - 3.16T + 43T^{2} \) |
| 47 | \( 1 + 9.89iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 13.4iT - 61T^{2} \) |
| 67 | \( 1 - 15.8T + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 - 15.5iT - 83T^{2} \) |
| 89 | \( 1 + 17.8iT - 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−10.15753416837018000057042160095, −9.709278380018870779489779819440, −8.423227978909403466924876838121, −7.78600925984069547620089997672, −6.80624283137444811309504573820, −5.99572966261366046677481883810, −4.66496397812076757902402182635, −3.74947851049353790014571509200, −1.98240849387952107145380553813, −0.67561751341928341798260270317,
2.29665644802958783081733835675, 3.41563597889183892758134941924, 4.36042514419845706968061955824, 5.70131989134314967577619354362, 6.45940279819211585448235441510, 7.50043012898668020424595708690, 8.603040950078407722285760719092, 9.679268621874590450799112854717, 10.09257098770096965741853736130, 10.91936781718765626626305044769