| L(s) = 1 | − 4.37·2-s + 11.1·4-s − 4.62·5-s + 12.1·7-s − 13.6·8-s + 20.2·10-s − 10.0·11-s − 48.5·13-s − 52.9·14-s − 29.3·16-s − 75.3·17-s − 116.·19-s − 51.4·20-s + 43.8·22-s + 38.0·23-s − 103.·25-s + 212.·26-s + 134.·28-s + 22.6·29-s + 30.1·31-s + 237.·32-s + 329.·34-s − 56.0·35-s + 130.·37-s + 507.·38-s + 63.0·40-s − 347.·41-s + ⋯ |
| L(s) = 1 | − 1.54·2-s + 1.38·4-s − 0.413·5-s + 0.654·7-s − 0.602·8-s + 0.639·10-s − 0.274·11-s − 1.03·13-s − 1.01·14-s − 0.458·16-s − 1.07·17-s − 1.40·19-s − 0.575·20-s + 0.424·22-s + 0.345·23-s − 0.828·25-s + 1.60·26-s + 0.909·28-s + 0.144·29-s + 0.174·31-s + 1.31·32-s + 1.66·34-s − 0.270·35-s + 0.578·37-s + 2.16·38-s + 0.249·40-s − 1.32·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 2 | \( 1 + 4.37T + 8T^{2} \) |
| 5 | \( 1 + 4.62T + 125T^{2} \) |
| 7 | \( 1 - 12.1T + 343T^{2} \) |
| 11 | \( 1 + 10.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 48.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 75.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 116.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 38.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 22.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 30.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 130.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 347.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 26.7T + 7.95e4T^{2} \) |
| 47 | \( 1 + 460.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 438.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 8.36T + 2.05e5T^{2} \) |
| 61 | \( 1 + 82.0T + 2.26e5T^{2} \) |
| 67 | \( 1 - 683.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.09e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 470.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 486.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 99.1T + 5.71e5T^{2} \) |
| 89 | \( 1 + 8.80T + 7.04e5T^{2} \) |
| 97 | \( 1 - 660.T + 9.12e5T^{2} \) |
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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
−13.22108481913971931631577836250, −11.77377589333832168646177016946, −10.89513760914469573537954295692, −9.889093309361191149801700439886, −8.674933380711729500907866632007, −7.87727581625584982046037480331, −6.73927698132108676157735784772, −4.63075146327798786792425699523, −2.10838781592008252593949288477, 0,
2.10838781592008252593949288477, 4.63075146327798786792425699523, 6.73927698132108676157735784772, 7.87727581625584982046037480331, 8.674933380711729500907866632007, 9.889093309361191149801700439886, 10.89513760914469573537954295692, 11.77377589333832168646177016946, 13.22108481913971931631577836250
