L(s) = 1 | + (−0.938 + 2.84i)3-s + (−4.88 + 1.05i)5-s − 6.81i·7-s + (−7.23 − 5.34i)9-s + 7.52i·11-s − 16.2i·13-s + (1.58 − 14.9i)15-s + 4.11·17-s + 7.86·19-s + (19.4 + 6.39i)21-s + 19.5·23-s + (22.7 − 10.2i)25-s + (22.0 − 15.6i)27-s + 55.8i·29-s − 43.4·31-s + ⋯ |
L(s) = 1 | + (−0.312 + 0.949i)3-s + (−0.977 + 0.210i)5-s − 0.973i·7-s + (−0.804 − 0.594i)9-s + 0.684i·11-s − 1.24i·13-s + (0.105 − 0.994i)15-s + 0.242·17-s + 0.413·19-s + (0.924 + 0.304i)21-s + 0.848·23-s + (0.911 − 0.411i)25-s + (0.815 − 0.578i)27-s + 1.92i·29-s − 1.40·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.105 - 0.994i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.105 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.039693553\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.039693553\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.938 - 2.84i)T \) |
| 5 | \( 1 + (4.88 - 1.05i)T \) |
good | 7 | \( 1 + 6.81iT - 49T^{2} \) |
| 11 | \( 1 - 7.52iT - 121T^{2} \) |
| 13 | \( 1 + 16.2iT - 169T^{2} \) |
| 17 | \( 1 - 4.11T + 289T^{2} \) |
| 19 | \( 1 - 7.86T + 361T^{2} \) |
| 23 | \( 1 - 19.5T + 529T^{2} \) |
| 29 | \( 1 - 55.8iT - 841T^{2} \) |
| 31 | \( 1 + 43.4T + 961T^{2} \) |
| 37 | \( 1 - 31.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 51.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 51.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 61.7T + 2.20e3T^{2} \) |
| 53 | \( 1 + 82.7T + 2.80e3T^{2} \) |
| 59 | \( 1 + 97.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 4.13T + 3.72e3T^{2} \) |
| 67 | \( 1 - 63.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 40.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 78.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 51.0T + 6.24e3T^{2} \) |
| 83 | \( 1 - 2.72T + 6.88e3T^{2} \) |
| 89 | \( 1 - 70.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 3.44iT - 9.40e3T^{2} \) |
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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
−10.24516542705996369227740728316, −9.318343226488752977268947726023, −8.348903507200751426279935089083, −7.45136400992640927741367126633, −6.81135071742206449663325587829, −5.39532911331412417517956374522, −4.73045091602286612294242206830, −3.68890996907171780056523641592, −3.11478179494792366378091777672, −0.871581165235597211216231135140,
0.48387955205153290836873445307, 1.91976588082227779855619736740, 3.08418435030705835494416960940, 4.32575318967308502206870922572, 5.47235051087135326186824607448, 6.18021159868819924034125789875, 7.23706886004710246926063402983, 7.82387995001598322993784280959, 8.823635920048403690417365257067, 9.235821911338727524445759715507