| L(s) = 1 | + 4.74·2-s − 3·3-s + 14.4·4-s + 4.51·5-s − 14.2·6-s − 7.48·7-s + 30.7·8-s + 9·9-s + 21.4·10-s − 66.8·11-s − 43.4·12-s − 13·13-s − 35.4·14-s − 13.5·15-s + 29.8·16-s + 96.9·17-s + 42.6·18-s + 31.4·19-s + 65.4·20-s + 22.4·21-s − 317.·22-s + 183.·23-s − 92.2·24-s − 104.·25-s − 61.6·26-s − 27·27-s − 108.·28-s + ⋯ |
| L(s) = 1 | + 1.67·2-s − 0.577·3-s + 1.81·4-s + 0.403·5-s − 0.967·6-s − 0.404·7-s + 1.35·8-s + 0.333·9-s + 0.677·10-s − 1.83·11-s − 1.04·12-s − 0.277·13-s − 0.677·14-s − 0.233·15-s + 0.467·16-s + 1.38·17-s + 0.558·18-s + 0.380·19-s + 0.731·20-s + 0.233·21-s − 3.07·22-s + 1.66·23-s − 0.784·24-s − 0.836·25-s − 0.464·26-s − 0.192·27-s − 0.731·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.455179940\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.455179940\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 13 | \( 1 + 13T \) |
| good | 2 | \( 1 - 4.74T + 8T^{2} \) |
| 5 | \( 1 - 4.51T + 125T^{2} \) |
| 7 | \( 1 + 7.48T + 343T^{2} \) |
| 11 | \( 1 + 66.8T + 1.33e3T^{2} \) |
| 17 | \( 1 - 96.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 31.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 183.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 112.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 77.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 54.7T + 5.06e4T^{2} \) |
| 41 | \( 1 - 451.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 113.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 42.2T + 1.03e5T^{2} \) |
| 53 | \( 1 + 530.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 219.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 822.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 872.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 100.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 165.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 545.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 454.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 230.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.08e3T + 9.12e5T^{2} \) |
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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
−15.59444456931606781246486698826, −14.39854807466564035287813392804, −13.16394236659639746642859752325, −12.61014534318416531167169799484, −11.25621080408741270288584986803, −10.00945851307149139236838779151, −7.43059213178317394275594944919, −5.85648758883860065584668602272, −4.99204993514547714957112126026, −2.95880254359165764848938604180,
2.95880254359165764848938604180, 4.99204993514547714957112126026, 5.85648758883860065584668602272, 7.43059213178317394275594944919, 10.00945851307149139236838779151, 11.25621080408741270288584986803, 12.61014534318416531167169799484, 13.16394236659639746642859752325, 14.39854807466564035287813392804, 15.59444456931606781246486698826
