| L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s + 11-s + 13-s + 16-s + 5·19-s − 20-s − 22-s + 23-s + 25-s − 26-s + 5·29-s + 7·31-s − 32-s + 3·37-s − 5·38-s + 40-s − 10·41-s + 7·43-s + 44-s − 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 + 0.301·11-s + 0.277·13-s + 1/4·16-s + 1.14·19-s − 0.223·20-s − 0.213·22-s + 0.208·23-s + 1/5·25-s − 0.196·26-s + 0.928·29-s + 1.25·31-s − 0.176·32-s + 0.493·37-s − 0.811·38-s + 0.158·40-s − 1.56·41-s + 1.06·43-s + 0.150·44-s − 0.147·46-s − 0.875·47-s − 49-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−12.79835455864360, −12.15442500294911, −11.80137994532273, −11.38907843035026, −11.15129430641975, −10.31440464317898, −10.15895022541110, −9.603098466774242, −9.188868432470611, −8.576071015802383, −8.328267247685178, −7.691336730012503, −7.501119581806395, −6.680895210527312, −6.496546227766178, −5.981656108532398, −5.187229970313971, −4.861134644924136, −4.259532305816956, −3.576926321730158, −3.111067634892286, −2.707188894176279, −1.885238150232655, −1.246756050338871, −0.8211153550058275, 0,
0.8211153550058275, 1.246756050338871, 1.885238150232655, 2.707188894176279, 3.111067634892286, 3.576926321730158, 4.259532305816956, 4.861134644924136, 5.187229970313971, 5.981656108532398, 6.496546227766178, 6.680895210527312, 7.501119581806395, 7.691336730012503, 8.328267247685178, 8.576071015802383, 9.188868432470611, 9.603098466774242, 10.15895022541110, 10.31440464317898, 11.15129430641975, 11.38907843035026, 11.80137994532273, 12.15442500294911, 12.79835455864360
