| 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
−14.37333329415361368995983884921, −13.49409411813013064037570130308, −12.44538507448031793314842320118, −11.02984248980745870084977485245, −9.841903954161407146509914296003, −9.087575197525150666800088693080, −6.93351743318834851212330700639, −6.20248517156742990520095981547, −4.73362069766530667139199063084, −1.95996050183106650780753721082,
3.08504180861655191323379640648, 4.73108416552243917878490880623, 6.14448455106215674371962373168, 7.86037312448301623642349171980, 9.250904419519294941988616175233, 9.923245884919107899280989199875, 11.56779758300431173700494865692, 12.48024315650644570217929376183, 13.62599577495890836643506762083, 14.41426324154526380320910611313
