L(s) = 1 | + 2-s + 4-s − 4·5-s + 8-s − 4·10-s + 4·11-s + 4·13-s + 16-s + 4·19-s − 4·20-s + 4·22-s + 11·25-s + 4·26-s − 2·29-s + 8·31-s + 32-s − 6·37-s + 4·38-s − 4·40-s + 4·43-s + 4·44-s + 8·47-s + 11·50-s + 4·52-s + 10·53-s − 16·55-s − 2·58-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.78·5-s + 0.353·8-s − 1.26·10-s + 1.20·11-s + 1.10·13-s + 1/4·16-s + 0.917·19-s − 0.894·20-s + 0.852·22-s + 11/5·25-s + 0.784·26-s − 0.371·29-s + 1.43·31-s + 0.176·32-s − 0.986·37-s + 0.648·38-s − 0.632·40-s + 0.609·43-s + 0.603·44-s + 1.16·47-s + 1.55·50-s + 0.554·52-s + 1.37·53-s − 2.15·55-s − 0.262·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.008060096\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.008060096\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.45018176958530837111076461590, −9.083181343332229084801124216818, −8.367002712496184399269342005462, −7.45554887409285403711020085491, −6.77970713278050838983490323162, −5.74587009674352948543449406475, −4.45230932233748260088756468320, −3.87915627242789051687615484994, −3.09526891884004259957229060750, −1.10984321000633935751921154525,
1.10984321000633935751921154525, 3.09526891884004259957229060750, 3.87915627242789051687615484994, 4.45230932233748260088756468320, 5.74587009674352948543449406475, 6.77970713278050838983490323162, 7.45554887409285403711020085491, 8.367002712496184399269342005462, 9.083181343332229084801124216818, 10.45018176958530837111076461590