| L(s) = 1 | + (−0.707 + 1.22i)2-s + (−0.522 − 2.95i)3-s + (−0.999 − 1.73i)4-s − 5.63·5-s + (3.98 + 1.44i)6-s + (−10.8 + 6.28i)7-s + 2.82·8-s + (−8.45 + 3.08i)9-s + (3.98 − 6.89i)10-s + (3.34 − 5.78i)11-s + (−4.59 + 3.85i)12-s + (3.64 − 12.4i)13-s − 17.7i·14-s + (2.94 + 16.6i)15-s + (−2.00 + 3.46i)16-s + (13.5 − 7.84i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.174 − 0.984i)3-s + (−0.249 − 0.433i)4-s − 1.12·5-s + (0.664 + 0.241i)6-s + (−1.55 + 0.897i)7-s + 0.353·8-s + (−0.939 + 0.343i)9-s + (0.398 − 0.689i)10-s + (0.303 − 0.526i)11-s + (−0.382 + 0.321i)12-s + (0.280 − 0.959i)13-s − 1.26i·14-s + (0.196 + 1.10i)15-s + (−0.125 + 0.216i)16-s + (0.799 − 0.461i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.910 + 0.414i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.910 + 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0347053 - 0.159991i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0347053 - 0.159991i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.707 - 1.22i)T \) |
| 3 | \( 1 + (0.522 + 2.95i)T \) |
| 13 | \( 1 + (-3.64 + 12.4i)T \) |
| good | 5 | \( 1 + 5.63T + 25T^{2} \) |
| 7 | \( 1 + (10.8 - 6.28i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-3.34 + 5.78i)T + (-60.5 - 104. i)T^{2} \) |
| 17 | \( 1 + (-13.5 + 7.84i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (8.05 - 4.65i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (31.2 + 18.0i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-10.4 - 6.02i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 - 28.5iT - 961T^{2} \) |
| 37 | \( 1 + (-0.0845 - 0.0488i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (5.07 - 8.78i)T + (-840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (34.2 + 59.3i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + 57.8T + 2.20e3T^{2} \) |
| 53 | \( 1 - 57.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-19.9 - 34.5i)T + (-1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-5.10 - 8.83i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (66.1 + 38.1i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + (24.1 + 41.8i)T + (-2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 - 8.56iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 37.3T + 6.24e3T^{2} \) |
| 83 | \( 1 - 52.7T + 6.88e3T^{2} \) |
| 89 | \( 1 + (-49.4 + 85.6i)T + (-3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-43.0 + 24.8i)T + (4.70e3 - 8.14e3i)T^{2} \) |
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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
−13.69469289155460471720454627193, −12.49679434997861375453988299946, −11.92754782073253550551234151925, −10.31629201640850888207925616227, −8.765539181869679067760547893776, −7.924832113345618321804986633925, −6.64124476607829531001223071568, −5.72947746448563830780215536804, −3.23857460588838303970321246495, −0.15283845295118365585273302976,
3.54894254592348638844659525962, 4.17818513704413198824002759592, 6.49862315357884329384525666624, 7.996768409999354385915299164534, 9.513797595829614765471008864177, 10.08681444478112786245256667763, 11.34397951006850320078163022907, 12.15769571668907426983491650827, 13.45011863774985782827455680397, 14.80858485875580225630100079391
