| L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s + 4·11-s + 2·13-s + 16-s − 4·17-s − 20-s − 4·22-s − 6·23-s + 25-s − 2·26-s + 10·29-s − 2·31-s − 32-s + 4·34-s + 2·37-s + 40-s + 6·41-s − 8·43-s + 4·44-s + 6·46-s + 6·47-s − 7·49-s − 50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s + 1.20·11-s + 0.554·13-s + 1/4·16-s − 0.970·17-s − 0.223·20-s − 0.852·22-s − 1.25·23-s + 1/5·25-s − 0.392·26-s + 1.85·29-s − 0.359·31-s − 0.176·32-s + 0.685·34-s + 0.328·37-s + 0.158·40-s + 0.937·41-s − 1.21·43-s + 0.603·44-s + 0.884·46-s + 0.875·47-s − 49-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−15.47637095184733, −14.76493640006016, −14.36224535928180, −13.78508795846587, −13.21485987121471, −12.45623383861441, −12.01551100553167, −11.55002498474094, −11.05361858324950, −10.52386107036086, −9.858243647343472, −9.373552870652719, −8.771084091938541, −8.300458109065040, −7.889586967510718, −7.013770067983555, −6.560390838627732, −6.199352207657800, −5.359211887543156, −4.383862758191492, −4.085803843352483, −3.275141664761144, −2.511724800252929, −1.676193820674592, −0.9916938879566991, 0,
0.9916938879566991, 1.676193820674592, 2.511724800252929, 3.275141664761144, 4.085803843352483, 4.383862758191492, 5.359211887543156, 6.199352207657800, 6.560390838627732, 7.013770067983555, 7.889586967510718, 8.300458109065040, 8.771084091938541, 9.373552870652719, 9.858243647343472, 10.52386107036086, 11.05361858324950, 11.55002498474094, 12.01551100553167, 12.45623383861441, 13.21485987121471, 13.78508795846587, 14.36224535928180, 14.76493640006016, 15.47637095184733
