| L(s) = 1 | + 1.50·2-s + 6.46·3-s − 5.74·4-s − 0.662·5-s + 9.70·6-s + 11.3·7-s − 20.6·8-s + 14.8·9-s − 0.993·10-s + 15.0·11-s − 37.1·12-s − 73.5·13-s + 17.0·14-s − 4.28·15-s + 15.0·16-s − 89.5·17-s + 22.2·18-s − 47.9·19-s + 3.80·20-s + 73.2·21-s + 22.5·22-s − 23·23-s − 133.·24-s − 124.·25-s − 110.·26-s − 78.7·27-s − 65.1·28-s + ⋯ |
| L(s) = 1 | + 0.530·2-s + 1.24·3-s − 0.718·4-s − 0.0592·5-s + 0.660·6-s + 0.611·7-s − 0.911·8-s + 0.548·9-s − 0.0314·10-s + 0.411·11-s − 0.894·12-s − 1.56·13-s + 0.324·14-s − 0.0737·15-s + 0.234·16-s − 1.27·17-s + 0.291·18-s − 0.579·19-s + 0.0425·20-s + 0.761·21-s + 0.218·22-s − 0.208·23-s − 1.13·24-s − 0.996·25-s − 0.832·26-s − 0.561·27-s − 0.439·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 23 | \( 1 + 23T \) |
| 29 | \( 1 - 29T \) |
| good | 2 | \( 1 - 1.50T + 8T^{2} \) |
| 3 | \( 1 - 6.46T + 27T^{2} \) |
| 5 | \( 1 + 0.662T + 125T^{2} \) |
| 7 | \( 1 - 11.3T + 343T^{2} \) |
| 11 | \( 1 - 15.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 73.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 89.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 47.9T + 6.85e3T^{2} \) |
| 31 | \( 1 - 136.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 65.6T + 5.06e4T^{2} \) |
| 41 | \( 1 + 287.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 9.56T + 7.95e4T^{2} \) |
| 47 | \( 1 - 404.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 601.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 496.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 751.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 91.2T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.01e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 513.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 369.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.24e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.14e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 390.T + 9.12e5T^{2} \) |
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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
−9.449256653099077197343162082407, −8.806161441678010674819830329749, −8.126444236162027286895054199814, −7.21756333509329716921804199882, −5.94081025802270651336406345439, −4.68472180122421652625585560494, −4.17901837138332678757460851153, −2.94612381632266085649810135259, −2.01232596359758944809265185634, 0,
2.01232596359758944809265185634, 2.94612381632266085649810135259, 4.17901837138332678757460851153, 4.68472180122421652625585560494, 5.94081025802270651336406345439, 7.21756333509329716921804199882, 8.126444236162027286895054199814, 8.806161441678010674819830329749, 9.449256653099077197343162082407
