L(s) = 1 | + 2-s + 4-s − 7-s + 8-s + 3·11-s + 13-s − 14-s + 16-s − 4·19-s + 3·22-s − 3·23-s + 26-s − 28-s − 6·29-s − 4·31-s + 32-s + 37-s − 4·38-s − 6·41-s − 8·43-s + 3·44-s − 3·46-s − 3·47-s + 49-s + 52-s − 12·53-s − 56-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s + 0.904·11-s + 0.277·13-s − 0.267·14-s + 1/4·16-s − 0.917·19-s + 0.639·22-s − 0.625·23-s + 0.196·26-s − 0.188·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s + 0.164·37-s − 0.648·38-s − 0.937·41-s − 1.21·43-s + 0.452·44-s − 0.442·46-s − 0.437·47-s + 1/7·49-s + 0.138·52-s − 1.64·53-s − 0.133·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−7.21027100117232556904869349479, −6.38761919067650322337188387993, −6.20857211786784885511667761256, −5.24050344526734225747907548381, −4.54836500325770325208403754501, −3.71224490788179778837895848773, −3.35746737931091054271286775594, −2.16715827260138251452903243219, −1.50361312920855195002421587936, 0,
1.50361312920855195002421587936, 2.16715827260138251452903243219, 3.35746737931091054271286775594, 3.71224490788179778837895848773, 4.54836500325770325208403754501, 5.24050344526734225747907548381, 6.20857211786784885511667761256, 6.38761919067650322337188387993, 7.21027100117232556904869349479