| L(s) = 1 | + (1.22 − 0.707i)2-s + (0.866 − 1.5i)3-s + (0.999 − 1.73i)4-s − 2.44i·6-s + (−5.76 + 3.97i)7-s − 2.82i·8-s + (−1.5 − 2.59i)9-s + (−1.64 + 2.85i)11-s + (−1.73 − 2.99i)12-s − 7.72·13-s + (−4.24 + 8.94i)14-s + (−2.00 − 3.46i)16-s + (−6.30 + 10.9i)17-s + (−3.67 − 2.12i)18-s + (−1.54 + 0.890i)19-s + ⋯ |
| L(s) = 1 | + (0.612 − 0.353i)2-s + (0.288 − 0.5i)3-s + (0.249 − 0.433i)4-s − 0.408i·6-s + (−0.823 + 0.567i)7-s − 0.353i·8-s + (−0.166 − 0.288i)9-s + (−0.149 + 0.259i)11-s + (−0.144 − 0.249i)12-s − 0.594·13-s + (−0.303 + 0.638i)14-s + (−0.125 − 0.216i)16-s + (−0.371 + 0.642i)17-s + (−0.204 − 0.117i)18-s + (−0.0812 + 0.0468i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.263 - 0.964i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.263 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6073772129\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6073772129\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.22 + 0.707i)T \) |
| 3 | \( 1 + (-0.866 + 1.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (5.76 - 3.97i)T \) |
| good | 11 | \( 1 + (1.64 - 2.85i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + 7.72T + 169T^{2} \) |
| 17 | \( 1 + (6.30 - 10.9i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (1.54 - 0.890i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (5.85 - 3.37i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 39.2T + 841T^{2} \) |
| 31 | \( 1 + (-9.46 - 5.46i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (29.7 - 17.1i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 - 18.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 77.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (6.44 + 11.1i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-44.7 - 25.8i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (97.7 + 56.4i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (22.7 - 13.1i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (17.2 + 9.95i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 87.4T + 5.04e3T^{2} \) |
| 73 | \( 1 + (30.4 - 52.7i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (17.7 + 30.8i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 46.6T + 6.88e3T^{2} \) |
| 89 | \( 1 + (-47.4 + 27.4i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 45.7T + 9.40e3T^{2} \) |
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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
−9.890501709396104604046076912052, −9.297479593734469042091741748742, −8.296895455442248881718275168307, −7.32961905286978846655821876060, −6.46140000745652591228624004970, −5.76573628517412428397977244999, −4.69099683721184077458260267893, −3.55097718808011082923191827680, −2.66532630278024444160975907387, −1.68221088019155216394889385055,
0.13169094320852044026516343736, 2.29218306631369685791643646717, 3.35550441639394411237439042369, 4.10681921187096521861003526433, 5.09587047659815931859610907064, 5.96999791850274211116405308910, 7.02146040009309491173935804668, 7.56137477166990058522824415042, 8.728726781438389339133122003972, 9.438868097228276522253269155525
