| L(s) = 1 | + 2-s + 4-s + 4·7-s + 8-s − 6·11-s − 4·13-s + 4·14-s + 16-s + 6·17-s − 6·22-s + 6·23-s − 4·26-s + 4·28-s + 2·29-s + 32-s + 6·34-s − 8·37-s + 10·41-s + 4·43-s − 6·44-s + 6·46-s − 2·47-s + 9·49-s − 4·52-s − 10·53-s + 4·56-s + 2·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.51·7-s + 0.353·8-s − 1.80·11-s − 1.10·13-s + 1.06·14-s + 1/4·16-s + 1.45·17-s − 1.27·22-s + 1.25·23-s − 0.784·26-s + 0.755·28-s + 0.371·29-s + 0.176·32-s + 1.02·34-s − 1.31·37-s + 1.56·41-s + 0.609·43-s − 0.904·44-s + 0.884·46-s − 0.291·47-s + 9/7·49-s − 0.554·52-s − 1.37·53-s + 0.534·56-s + 0.262·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−13.58958214522438, −12.90201073033472, −12.60250400790024, −12.12169194694095, −11.77773710571374, −10.92724253666885, −10.83365011442319, −10.44202278568791, −9.708014844869407, −9.311906705764995, −8.441318197562860, −8.044904544074796, −7.608266623679952, −7.389491889724642, −6.741285481156611, −5.819619272470075, −5.466909445423959, −5.040351610363922, −4.725244608568273, −4.203678551500717, −3.200630825011197, −2.892582412253408, −2.327082782493521, −1.630177611195930, −1.016228742583560, 0,
1.016228742583560, 1.630177611195930, 2.327082782493521, 2.892582412253408, 3.200630825011197, 4.203678551500717, 4.725244608568273, 5.040351610363922, 5.466909445423959, 5.819619272470075, 6.741285481156611, 7.389491889724642, 7.608266623679952, 8.044904544074796, 8.441318197562860, 9.311906705764995, 9.708014844869407, 10.44202278568791, 10.83365011442319, 10.92724253666885, 11.77773710571374, 12.12169194694095, 12.60250400790024, 12.90201073033472, 13.58958214522438
