| L(s) = 1 | + (0.114 − 0.0832i)2-s + (−0.309 + 0.951i)3-s + (−0.611 + 1.88i)4-s + (2.14 − 0.617i)5-s + (0.0437 + 0.134i)6-s − 0.858·7-s + (0.174 + 0.536i)8-s + (−0.809 − 0.587i)9-s + (0.194 − 0.249i)10-s + (2.97 − 2.16i)11-s + (−1.60 − 1.16i)12-s + (−3.70 − 2.69i)13-s + (−0.0983 + 0.0714i)14-s + (−0.0763 + 2.23i)15-s + (−3.13 − 2.28i)16-s + (−1.63 − 5.04i)17-s + ⋯ |
| L(s) = 1 | + (0.0810 − 0.0588i)2-s + (−0.178 + 0.549i)3-s + (−0.305 + 0.941i)4-s + (0.961 − 0.276i)5-s + (0.0178 + 0.0550i)6-s − 0.324·7-s + (0.0616 + 0.189i)8-s + (−0.269 − 0.195i)9-s + (0.0616 − 0.0789i)10-s + (0.897 − 0.652i)11-s + (−0.462 − 0.335i)12-s + (−1.02 − 0.746i)13-s + (−0.0262 + 0.0191i)14-s + (−0.0197 + 0.577i)15-s + (−0.784 − 0.570i)16-s + (−0.397 − 1.22i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.773 - 0.633i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.773 - 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.897060 + 0.320190i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.897060 + 0.320190i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.309 - 0.951i)T \) |
| 5 | \( 1 + (-2.14 + 0.617i)T \) |
| good | 2 | \( 1 + (-0.114 + 0.0832i)T + (0.618 - 1.90i)T^{2} \) |
| 7 | \( 1 + 0.858T + 7T^{2} \) |
| 11 | \( 1 + (-2.97 + 2.16i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (3.70 + 2.69i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (1.63 + 5.04i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-1.96 - 6.05i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (2.76 - 2.01i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-1.15 + 3.55i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-0.387 - 1.19i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-6.02 - 4.37i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-2.04 - 1.48i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 3.37T + 43T^{2} \) |
| 47 | \( 1 + (2.62 - 8.08i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (0.725 - 2.23i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (10.6 + 7.71i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (8.37 - 6.08i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-1.03 - 3.18i)T + (-54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (1.33 - 4.11i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-7.34 + 5.33i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (1.00 - 3.08i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (2.28 + 7.03i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-12.5 + 9.10i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (3.10 - 9.54i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−14.41426324154526380320910611313, −13.62599577495890836643506762083, −12.48024315650644570217929376183, −11.56779758300431173700494865692, −9.923245884919107899280989199875, −9.250904419519294941988616175233, −7.86037312448301623642349171980, −6.14448455106215674371962373168, −4.73108416552243917878490880623, −3.08504180861655191323379640648,
1.95996050183106650780753721082, 4.73362069766530667139199063084, 6.20248517156742990520095981547, 6.93351743318834851212330700639, 9.087575197525150666800088693080, 9.841903954161407146509914296003, 11.02984248980745870084977485245, 12.44538507448031793314842320118, 13.49409411813013064037570130308, 14.37333329415361368995983884921
