| L(s) = 1 | − 3.16·2-s + 2.00·4-s − 15·7-s + 18.9·8-s + 63.2·11-s − 35·13-s + 47.4·14-s − 76·16-s − 88.5·17-s + 91·19-s − 200·22-s + 113.·23-s + 110.·26-s − 30.0·28-s − 63.2·29-s − 147·31-s + 88.5·32-s + 280·34-s − 370·37-s − 287.·38-s − 442.·41-s − 335·43-s + 126.·44-s − 360·46-s + 177.·47-s − 118·49-s − 70.0·52-s + ⋯ |
| L(s) = 1 | − 1.11·2-s + 0.250·4-s − 0.809·7-s + 0.838·8-s + 1.73·11-s − 0.746·13-s + 0.905·14-s − 1.18·16-s − 1.26·17-s + 1.09·19-s − 1.93·22-s + 1.03·23-s + 0.834·26-s − 0.202·28-s − 0.404·29-s − 0.851·31-s + 0.489·32-s + 1.41·34-s − 1.64·37-s − 1.22·38-s − 1.68·41-s − 1.18·43-s + 0.433·44-s − 1.15·46-s + 0.549·47-s − 0.344·49-s − 0.186·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 3.16T + 8T^{2} \) |
| 7 | \( 1 + 15T + 343T^{2} \) |
| 11 | \( 1 - 63.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 35T + 2.19e3T^{2} \) |
| 17 | \( 1 + 88.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 91T + 6.85e3T^{2} \) |
| 23 | \( 1 - 113.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 63.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 147T + 2.97e4T^{2} \) |
| 37 | \( 1 + 370T + 5.06e4T^{2} \) |
| 41 | \( 1 + 442.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 335T + 7.95e4T^{2} \) |
| 47 | \( 1 - 177.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 88.5T + 1.48e5T^{2} \) |
| 59 | \( 1 + 885.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 427T + 2.26e5T^{2} \) |
| 67 | \( 1 + 15T + 3.00e5T^{2} \) |
| 71 | \( 1 + 63.2T + 3.57e5T^{2} \) |
| 73 | \( 1 - 70T + 3.89e5T^{2} \) |
| 79 | \( 1 + 876T + 4.93e5T^{2} \) |
| 83 | \( 1 - 531.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.08e3T + 9.12e5T^{2} \) |
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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
−11.15314918098509793744635311909, −10.01749505805454377297480946277, −9.257855734092674488226791836294, −8.746893228805357428096158941487, −7.23311097739319602963219167242, −6.65035031234585382071055308059, −4.91601179539327128908461726485, −3.51576106426752832375520331175, −1.59369093416360635764731345696, 0,
1.59369093416360635764731345696, 3.51576106426752832375520331175, 4.91601179539327128908461726485, 6.65035031234585382071055308059, 7.23311097739319602963219167242, 8.746893228805357428096158941487, 9.257855734092674488226791836294, 10.01749505805454377297480946277, 11.15314918098509793744635311909
