| L(s) = 1 | + 4.37·2-s + 11.1·4-s − 12.1·7-s + 13.6·8-s − 10.0·11-s + 48.5·13-s − 52.9·14-s − 29.3·16-s + 75.3·17-s − 116.·19-s − 43.8·22-s − 38.0·23-s + 212.·26-s − 134.·28-s + 22.6·29-s + 30.1·31-s − 237.·32-s + 329.·34-s − 130.·37-s − 507.·38-s − 347.·41-s − 26.7·43-s − 111.·44-s − 166.·46-s + 460.·47-s − 196.·49-s + 540.·52-s + ⋯ |
| L(s) = 1 | + 1.54·2-s + 1.38·4-s − 0.654·7-s + 0.602·8-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 − 0.424·22-s − 0.345·23-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 − 0.578·37-s − 2.16·38-s − 1.32·41-s − 0.0949·43-s − 0.381·44-s − 0.533·46-s + 1.43·47-s − 0.571·49-s + 1.44·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 4.37T + 8T^{2} \) |
| 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
−8.360307700159525156949225280743, −7.33752666147329640181808497179, −6.37312807530202333653737623495, −6.02979336777099538365126385811, −5.13820275269862156936450773136, −4.23388313276644903180198688664, −3.51220965268993638918695374399, −2.81394586325117450709485129656, −1.62187259801120991777345391730, 0,
1.62187259801120991777345391730, 2.81394586325117450709485129656, 3.51220965268993638918695374399, 4.23388313276644903180198688664, 5.13820275269862156936450773136, 6.02979336777099538365126385811, 6.37312807530202333653737623495, 7.33752666147329640181808497179, 8.360307700159525156949225280743
