L(s) = 1 | + 3-s − 5-s − 3·7-s + 9-s − 2·11-s − 2·13-s − 15-s − 4·17-s − 4·19-s − 3·21-s + 7·23-s − 4·25-s + 27-s − 8·29-s + 3·31-s − 2·33-s + 3·35-s − 3·37-s − 2·39-s + 41-s − 11·43-s − 45-s + 2·49-s − 4·51-s + 11·53-s + 2·55-s − 4·57-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.13·7-s + 1/3·9-s − 0.603·11-s − 0.554·13-s − 0.258·15-s − 0.970·17-s − 0.917·19-s − 0.654·21-s + 1.45·23-s − 4/5·25-s + 0.192·27-s − 1.48·29-s + 0.538·31-s − 0.348·33-s + 0.507·35-s − 0.493·37-s − 0.320·39-s + 0.156·41-s − 1.67·43-s − 0.149·45-s + 2/7·49-s − 0.560·51-s + 1.51·53-s + 0.269·55-s − 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 67 | \( 1 + T \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−9.750234589912995839555802068068, −9.010124849067987371032486144526, −8.199712549586293443059014780518, −7.19634459883027179171790840316, −6.58469246152879980363958305254, −5.31649357921964130503329570462, −4.18065040315259304688251616469, −3.23320972721445320648750162796, −2.22330826788334876611733342610, 0,
2.22330826788334876611733342610, 3.23320972721445320648750162796, 4.18065040315259304688251616469, 5.31649357921964130503329570462, 6.58469246152879980363958305254, 7.19634459883027179171790840316, 8.199712549586293443059014780518, 9.010124849067987371032486144526, 9.750234589912995839555802068068