| L(s) = 1 | − 4.31·2-s − 23.8·3-s − 13.3·4-s − 52.5·5-s + 102.·6-s − 174.·7-s + 195.·8-s + 325.·9-s + 226.·10-s − 447.·11-s + 319.·12-s + 669.·13-s + 754.·14-s + 1.25e3·15-s − 416.·16-s − 849.·17-s − 1.40e3·18-s − 1.28e3·19-s + 703.·20-s + 4.16e3·21-s + 1.93e3·22-s − 378.·23-s − 4.66e3·24-s − 359.·25-s − 2.88e3·26-s − 1.97e3·27-s + 2.33e3·28-s + ⋯ |
| L(s) = 1 | − 0.762·2-s − 1.52·3-s − 0.418·4-s − 0.940·5-s + 1.16·6-s − 1.34·7-s + 1.08·8-s + 1.34·9-s + 0.717·10-s − 1.11·11-s + 0.639·12-s + 1.09·13-s + 1.02·14-s + 1.43·15-s − 0.407·16-s − 0.713·17-s − 1.02·18-s − 0.819·19-s + 0.393·20-s + 2.06·21-s + 0.851·22-s − 0.149·23-s − 1.65·24-s − 0.115·25-s − 0.837·26-s − 0.520·27-s + 0.563·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.1263269118\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1263269118\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 - 1.84e3T \) |
| good | 2 | \( 1 + 4.31T + 32T^{2} \) |
| 3 | \( 1 + 23.8T + 243T^{2} \) |
| 5 | \( 1 + 52.5T + 3.12e3T^{2} \) |
| 7 | \( 1 + 174.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 447.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 669.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 849.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.28e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 378.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 765.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.09e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.90e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.28e4T + 1.15e8T^{2} \) |
| 47 | \( 1 - 2.67e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.00e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.24e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.84e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.70e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.45e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.80e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.32e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.80e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.03e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.75e4T + 8.58e9T^{2} \) |
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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
−15.73961968138714541544473240701, −13.32410699933760574850095596762, −12.50478319287197356409885994240, −11.03195992116345356465867149024, −10.34764918951052293666627721665, −8.797880452020365467018047324036, −7.25022158904520124227393199304, −5.81095157231506827861527316982, −4.11832708096723618849682078976, −0.35875812059184369204644146503,
0.35875812059184369204644146503, 4.11832708096723618849682078976, 5.81095157231506827861527316982, 7.25022158904520124227393199304, 8.797880452020365467018047324036, 10.34764918951052293666627721665, 11.03195992116345356465867149024, 12.50478319287197356409885994240, 13.32410699933760574850095596762, 15.73961968138714541544473240701
