| L(s) = 1 | − 9.18·2-s − 9·3-s + 52.2·4-s − 22.0·5-s + 82.6·6-s − 186.·8-s + 81·9-s + 202.·10-s + 416.·11-s − 470.·12-s − 797.·13-s + 198.·15-s + 37.3·16-s + 1.37e3·17-s − 743.·18-s − 2.31e3·19-s − 1.15e3·20-s − 3.82e3·22-s − 955.·23-s + 1.67e3·24-s − 2.63e3·25-s + 7.32e3·26-s − 729·27-s − 7.03e3·29-s − 1.82e3·30-s − 1.26e3·31-s + 5.61e3·32-s + ⋯ |
| L(s) = 1 | − 1.62·2-s − 0.577·3-s + 1.63·4-s − 0.394·5-s + 0.937·6-s − 1.02·8-s + 0.333·9-s + 0.640·10-s + 1.03·11-s − 0.943·12-s − 1.30·13-s + 0.227·15-s + 0.0364·16-s + 1.15·17-s − 0.541·18-s − 1.46·19-s − 0.645·20-s − 1.68·22-s − 0.376·23-s + 0.594·24-s − 0.844·25-s + 2.12·26-s − 0.192·27-s − 1.55·29-s − 0.369·30-s − 0.235·31-s + 0.970·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.4538217336\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4538217336\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 9T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 9.18T + 32T^{2} \) |
| 5 | \( 1 + 22.0T + 3.12e3T^{2} \) |
| 11 | \( 1 - 416.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 797.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.37e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.31e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 955.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.03e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.26e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 9.77e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 5.40e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.96e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.05e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.80e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 7.43e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.49e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.58e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.81e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.91e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.76e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.03e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.44e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 7.93e4T + 8.58e9T^{2} \) |
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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
−11.78778742558576801908053796560, −10.96553251873265945790879391251, −9.911687019331489087886981059640, −9.226895724240787243027349758834, −7.920021477549789167324028620144, −7.19223440578622148344131217763, −5.95581419629759966610896774074, −4.16943160633210232698693632090, −2.03985272499899840589554176227, −0.57327568500100226838185803414,
0.57327568500100226838185803414, 2.03985272499899840589554176227, 4.16943160633210232698693632090, 5.95581419629759966610896774074, 7.19223440578622148344131217763, 7.920021477549789167324028620144, 9.226895724240787243027349758834, 9.911687019331489087886981059640, 10.96553251873265945790879391251, 11.78778742558576801908053796560
