| L(s) = 1 | + 2.41·2-s + 3·3-s − 2.17·4-s − 19.8·5-s + 7.24·6-s − 24.5·8-s + 9·9-s − 48.0·10-s + 23.9·11-s − 6.51·12-s − 87.3·13-s − 59.6·15-s − 41.9·16-s + 5.63·17-s + 21.7·18-s − 64.8·19-s + 43.2·20-s + 57.7·22-s − 25.5·23-s − 73.6·24-s + 270.·25-s − 210.·26-s + 27·27-s + 60.3·29-s − 144.·30-s + 122.·31-s + 95.2·32-s + ⋯ |
| L(s) = 1 | + 0.853·2-s + 0.577·3-s − 0.271·4-s − 1.77·5-s + 0.492·6-s − 1.08·8-s + 0.333·9-s − 1.51·10-s + 0.656·11-s − 0.156·12-s − 1.86·13-s − 1.02·15-s − 0.654·16-s + 0.0804·17-s + 0.284·18-s − 0.783·19-s + 0.483·20-s + 0.560·22-s − 0.232·23-s − 0.626·24-s + 2.16·25-s − 1.59·26-s + 0.192·27-s + 0.386·29-s − 0.877·30-s + 0.710·31-s + 0.526·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 2.41T + 8T^{2} \) |
| 5 | \( 1 + 19.8T + 125T^{2} \) |
| 11 | \( 1 - 23.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 87.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 5.63T + 4.91e3T^{2} \) |
| 19 | \( 1 + 64.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 25.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 60.3T + 2.43e4T^{2} \) |
| 31 | \( 1 - 122.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 56.1T + 5.06e4T^{2} \) |
| 41 | \( 1 - 299.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 501.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 305.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 375.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 627.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 3.75T + 2.26e5T^{2} \) |
| 67 | \( 1 + 813.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 165.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 619.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 138.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 621.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 285.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 603.T + 9.12e5T^{2} \) |
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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
−12.18872343228824712740623164470, −11.59465265911880767224001042876, −9.964603338672840402562751343545, −8.761147090251000103918552675178, −7.85750044700660288309851316856, −6.75455036420968832171536398175, −4.81561375994549828243074126578, −4.10915585001137280153400116083, −2.96489447956054835972229221812, 0,
2.96489447956054835972229221812, 4.10915585001137280153400116083, 4.81561375994549828243074126578, 6.75455036420968832171536398175, 7.85750044700660288309851316856, 8.761147090251000103918552675178, 9.964603338672840402562751343545, 11.59465265911880767224001042876, 12.18872343228824712740623164470
