| L(s) = 1 | + (−1.37 + 1.45i)2-s + (−0.240 − 3.99i)4-s + 6.26i·5-s + 2.64i·7-s + (6.14 + 5.12i)8-s + (−9.12 − 8.59i)10-s − 9.80·11-s − 2.41i·13-s + (−3.85 − 3.62i)14-s + (−15.8 + 1.92i)16-s − 6.89·17-s + 2.77·19-s + (25.0 − 1.50i)20-s + (13.4 − 14.2i)22-s + 42.8i·23-s + ⋯ |
| L(s) = 1 | + (−0.685 + 0.728i)2-s + (−0.0602 − 0.998i)4-s + 1.25i·5-s + 0.377i·7-s + (0.768 + 0.640i)8-s + (−0.912 − 0.859i)10-s − 0.891·11-s − 0.185i·13-s + (−0.275 − 0.259i)14-s + (−0.992 + 0.120i)16-s − 0.405·17-s + 0.146·19-s + (1.25 − 0.0754i)20-s + (0.611 − 0.649i)22-s + 1.86i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.640 + 0.768i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.640 + 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3371713799\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3371713799\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.37 - 1.45i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
| good | 5 | \( 1 - 6.26iT - 25T^{2} \) |
| 11 | \( 1 + 9.80T + 121T^{2} \) |
| 13 | \( 1 + 2.41iT - 169T^{2} \) |
| 17 | \( 1 + 6.89T + 289T^{2} \) |
| 19 | \( 1 - 2.77T + 361T^{2} \) |
| 23 | \( 1 - 42.8iT - 529T^{2} \) |
| 29 | \( 1 + 37.3iT - 841T^{2} \) |
| 31 | \( 1 - 7.16iT - 961T^{2} \) |
| 37 | \( 1 - 0.202iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 63.5T + 1.68e3T^{2} \) |
| 43 | \( 1 + 35.3T + 1.84e3T^{2} \) |
| 47 | \( 1 + 37.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 54.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 104.T + 3.48e3T^{2} \) |
| 61 | \( 1 + 43.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 31.1T + 4.48e3T^{2} \) |
| 71 | \( 1 - 23.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 69.2T + 5.32e3T^{2} \) |
| 79 | \( 1 + 19.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 5.11T + 6.88e3T^{2} \) |
| 89 | \( 1 - 17.9T + 7.92e3T^{2} \) |
| 97 | \( 1 - 12.4T + 9.40e3T^{2} \) |
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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
−11.06446147563077877229570980185, −10.21250336415124798620668575975, −9.612234465843705767956807691060, −8.423687334154644812981983592851, −7.62038821369266009546455088869, −6.85200948861568230380566143716, −5.93435295354042987428244988864, −5.02298771865252992111564412300, −3.28966417462195649861543346916, −2.00881976168255784368267529787,
0.16809523766894250532757342178, 1.47638560343275401472868638865, 2.88600606985239967914356490728, 4.32393145935927069433673002497, 5.04422407593800350159966099196, 6.65865736043874047503346683883, 7.77795054373245470112538986499, 8.570920707248691064458588354096, 9.143804536516744795308456976263, 10.25529164541175768341317137258
