| L(s) = 1 | + (−1.67 + 2.49i)3-s − 12.7·7-s + (−3.41 − 8.32i)9-s + 12.6i·11-s − 7.44·13-s − 14.0i·17-s + 31.0·19-s + (21.3 − 31.8i)21-s − 7.50i·23-s + (26.4 + 5.40i)27-s − 15.7i·29-s − 20.4·31-s + (−31.4 − 21.1i)33-s + 12.9·37-s + (12.4 − 18.5i)39-s + ⋯ |
| L(s) = 1 | + (−0.557 + 0.830i)3-s − 1.82·7-s + (−0.379 − 0.925i)9-s + 1.14i·11-s − 0.572·13-s − 0.826i·17-s + 1.63·19-s + (1.01 − 1.51i)21-s − 0.326i·23-s + (0.979 + 0.200i)27-s − 0.542i·29-s − 0.660·31-s + (−0.953 − 0.639i)33-s + 0.349·37-s + (0.318 − 0.475i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.830 + 0.557i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.830 + 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7064003341\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7064003341\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.67 - 2.49i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 12.7T + 49T^{2} \) |
| 11 | \( 1 - 12.6iT - 121T^{2} \) |
| 13 | \( 1 + 7.44T + 169T^{2} \) |
| 17 | \( 1 + 14.0iT - 289T^{2} \) |
| 19 | \( 1 - 31.0T + 361T^{2} \) |
| 23 | \( 1 + 7.50iT - 529T^{2} \) |
| 29 | \( 1 + 15.7iT - 841T^{2} \) |
| 31 | \( 1 + 20.4T + 961T^{2} \) |
| 37 | \( 1 - 12.9T + 1.36e3T^{2} \) |
| 41 | \( 1 + 13.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 30.0T + 1.84e3T^{2} \) |
| 47 | \( 1 - 20.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 29.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 47.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 43.0T + 3.72e3T^{2} \) |
| 67 | \( 1 + 0.630T + 4.48e3T^{2} \) |
| 71 | \( 1 + 90.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 46.2T + 5.32e3T^{2} \) |
| 79 | \( 1 + 37.9T + 6.24e3T^{2} \) |
| 83 | \( 1 - 80.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 140. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 10.3T + 9.40e3T^{2} \) |
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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
−9.994530285109660539658575178448, −9.726517276523898867661269868623, −9.142355647805663397372021477699, −7.43928083190226190286488279918, −6.74883251015396954574271532978, −5.74239693969757984973061113903, −4.81538115096068949075957130531, −3.68694742045529228671074296636, −2.73216737326831889548166687605, −0.37191072568896962630656773025,
0.926904726034864310051876801979, 2.72463525864486073983309632707, 3.63455200113004658743663635513, 5.41663724806415534447120669105, 6.02554366628711547957841494063, 6.88863644444426874249503098449, 7.65966924872641727192047630883, 8.830950974884919796810701481166, 9.714557679833081753897402771509, 10.57191781083408001785634343586
