L(s) = 1 | + (1.23 − 2.13i)5-s + (2.32 + 4.02i)11-s + (−3.55 − 6.15i)13-s + (−2.25 + 3.90i)17-s + (−2.16 − 3.74i)19-s + (2.93 − 5.08i)23-s + (−0.527 − 0.912i)25-s + (−3.48 + 6.04i)29-s − 7.38·31-s + (0.363 + 0.629i)37-s + (0.136 + 0.236i)41-s + (2.41 − 4.18i)43-s − 3.67·47-s + (2.52 − 4.37i)53-s + 11.4·55-s + ⋯ |
L(s) = 1 | + (0.550 − 0.952i)5-s + (0.700 + 1.21i)11-s + (−0.985 − 1.70i)13-s + (−0.547 + 0.948i)17-s + (−0.496 − 0.859i)19-s + (0.611 − 1.05i)23-s + (−0.105 − 0.182i)25-s + (−0.647 + 1.12i)29-s − 1.32·31-s + (0.0597 + 0.103i)37-s + (0.0213 + 0.0369i)41-s + (0.368 − 0.638i)43-s − 0.535·47-s + (0.347 − 0.601i)53-s + 1.54·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00220i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00220i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5865505230\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5865505230\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-1.23 + 2.13i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.32 - 4.02i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (3.55 + 6.15i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (2.25 - 3.90i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.16 + 3.74i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.93 + 5.08i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.48 - 6.04i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 7.38T + 31T^{2} \) |
| 37 | \( 1 + (-0.363 - 0.629i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.136 - 0.236i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.41 + 4.18i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 3.67T + 47T^{2} \) |
| 53 | \( 1 + (-2.52 + 4.37i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 9.13T + 59T^{2} \) |
| 61 | \( 1 + 13.8T + 61T^{2} \) |
| 67 | \( 1 + 1.32T + 67T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 + (-2.16 + 3.74i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 - 6.43T + 79T^{2} \) |
| 83 | \( 1 + (-0.742 + 1.28i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-4.91 - 8.51i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.246 + 0.426i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−7.78342343911300891782794211547, −7.16602107797089503460245577205, −6.41255938522383061808684051124, −5.49431065259320480699889550085, −4.91927612397204428434611780335, −4.36295898458376890575103221043, −3.23596165351942359583179796562, −2.21529492538217191409352147242, −1.43282835606121779984374034315, −0.14156337789578506748212663450,
1.56032618737089086266146429405, 2.35759203086821261030771174535, 3.22855048982942273049590488579, 4.05550710448437613232956149834, 4.86948348289994356556636265995, 6.01345802208866638651980716847, 6.24289215292085418588007226036, 7.18784288209099497379559469830, 7.53402195929894640310653267825, 8.757767501449790950135765665903