| L(s) = 1 | + (−0.707 − 2.91i)3-s + (−2.82 − 4.12i)5-s − 5.83i·7-s + (−8 + 4.12i)9-s + 16.4i·11-s + (−10.0 + 11.1i)15-s + 11.3·17-s − 12·19-s + (−17 + 4.12i)21-s − 24.0·23-s + (−8.99 + 23.3i)25-s + (17.6 + 20.4i)27-s − 32·31-s + (48.0 − 11.6i)33-s + (−24.0 + 16.4i)35-s + ⋯ |
| L(s) = 1 | + (−0.235 − 0.971i)3-s + (−0.565 − 0.824i)5-s − 0.832i·7-s + (−0.888 + 0.458i)9-s + 1.49i·11-s + (−0.668 + 0.744i)15-s + 0.665·17-s − 0.631·19-s + (−0.809 + 0.196i)21-s − 1.04·23-s + (−0.359 + 0.932i)25-s + (0.654 + 0.755i)27-s − 1.03·31-s + (1.45 − 0.353i)33-s + (−0.686 + 0.471i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.668 - 0.744i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.668 - 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5762772609\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5762772609\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.707 + 2.91i)T \) |
| 5 | \( 1 + (2.82 + 4.12i)T \) |
| good | 7 | \( 1 + 5.83iT - 49T^{2} \) |
| 11 | \( 1 - 16.4iT - 121T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 - 11.3T + 289T^{2} \) |
| 19 | \( 1 + 12T + 361T^{2} \) |
| 23 | \( 1 + 24.0T + 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 + 32T + 961T^{2} \) |
| 37 | \( 1 - 23.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 57.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 40.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 35.3T + 2.20e3T^{2} \) |
| 53 | \( 1 - 67.8T + 2.80e3T^{2} \) |
| 59 | \( 1 - 16.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 16T + 3.72e3T^{2} \) |
| 67 | \( 1 + 5.83iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 116. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 72T + 6.24e3T^{2} \) |
| 83 | \( 1 + 43.8T + 6.88e3T^{2} \) |
| 89 | \( 1 - 65.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 163. iT - 9.40e3T^{2} \) |
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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
−9.995014401824615151089867054595, −8.981676425427663500262466535831, −7.995868250153502046371420971078, −7.48999306494675899980599372120, −6.76577432265066780296991903829, −5.61590164619443808746935352619, −4.65248764375503876294005136642, −3.78806781633621888736568104072, −2.13574007973778682430786326746, −1.09226487417911358152462033700,
0.21408149391259101036505499985, 2.50506970221317411018297371174, 3.45295824397742914573607108687, 4.15642802100913377457528180871, 5.70878939430692920827349816875, 5.84739706723480251254212806044, 7.16413336235925325128392857008, 8.340592512989470393052529014432, 8.781529062631607775616854446165, 9.853929249027517934431352463563
