L(s) = 1 | + (0.349 − 0.605i)5-s + (−2.02 + 1.70i)7-s + (−0.229 + 0.132i)11-s − 1.31i·13-s + (−1.86 − 3.22i)17-s + (−0.382 − 0.220i)19-s + (4.29 + 2.48i)23-s + (2.25 + 3.90i)25-s − 0.315i·29-s + (4.85 − 2.80i)31-s + (0.322 + 1.82i)35-s + (−0.351 + 0.608i)37-s + 10.7·41-s − 7.46·43-s + (3.50 − 6.06i)47-s + ⋯ |
L(s) = 1 | + (0.156 − 0.270i)5-s + (−0.765 + 0.643i)7-s + (−0.0692 + 0.0399i)11-s − 0.364i·13-s + (−0.452 − 0.783i)17-s + (−0.0877 − 0.0506i)19-s + (0.896 + 0.517i)23-s + (0.451 + 0.781i)25-s − 0.0585i·29-s + (0.872 − 0.503i)31-s + (0.0545 + 0.308i)35-s + (−0.0577 + 0.0999i)37-s + 1.68·41-s − 1.13·43-s + (0.510 − 0.884i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0484i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0484i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.590757317\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.590757317\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.02 - 1.70i)T \) |
good | 5 | \( 1 + (-0.349 + 0.605i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.229 - 0.132i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 1.31iT - 13T^{2} \) |
| 17 | \( 1 + (1.86 + 3.22i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.382 + 0.220i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.29 - 2.48i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 0.315iT - 29T^{2} \) |
| 31 | \( 1 + (-4.85 + 2.80i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.351 - 0.608i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 43 | \( 1 + 7.46T + 43T^{2} \) |
| 47 | \( 1 + (-3.50 + 6.06i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-8.51 + 4.91i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.73 - 11.6i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.89 - 2.82i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.97 - 5.14i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 13.4iT - 71T^{2} \) |
| 73 | \( 1 + (6.66 - 3.84i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.698 - 1.20i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 7.44T + 83T^{2} \) |
| 89 | \( 1 + (-5.59 + 9.68i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 10.6iT - 97T^{2} \) |
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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
−8.973193929166232333673410938871, −8.504876653930116417462774799159, −7.32476667249430187959993769338, −6.79728588961705384336421567266, −5.73759380190345488117357473812, −5.24719717042522998829111767678, −4.17472483970067175039282940091, −3.08484041909640907961767776983, −2.35550176469284964164408431567, −0.825836180540485080648224827087,
0.804081141609464637527659313183, 2.26690182003494364953443260648, 3.21784828992933651377594807705, 4.13705762019720857234454408936, 4.92397024675129136597237183521, 6.20560772087462035715810616461, 6.55274254786892261931145159787, 7.37154538617951372630236820724, 8.288641366588664939630321054603, 9.051021726886360543186195113999