| 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
−14.80858485875580225630100079391, −13.45011863774985782827455680397, −12.15769571668907426983491650827, −11.34397951006850320078163022907, −10.08681444478112786245256667763, −9.513797595829614765471008864177, −7.996768409999354385915299164534, −6.49862315357884329384525666624, −4.17818513704413198824002759592, −3.54894254592348638844659525962,
0.15283845295118365585273302976, 3.23857460588838303970321246495, 5.72947746448563830780215536804, 6.64124476607829531001223071568, 7.924832113345618321804986633925, 8.765539181869679067760547893776, 10.31629201640850888207925616227, 11.92754782073253550551234151925, 12.49679434997861375453988299946, 13.69469289155460471720454627193
