L(s) = 1 | + 3-s − 5-s + 2·7-s + 9-s + 2·11-s − 6·13-s − 15-s − 4·17-s + 2·21-s − 23-s + 25-s + 27-s + 2·29-s + 2·33-s − 2·35-s − 8·37-s − 6·39-s − 6·41-s + 4·43-s − 45-s − 3·49-s − 4·51-s + 6·53-s − 2·55-s − 8·61-s + 2·63-s + 6·65-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.603·11-s − 1.66·13-s − 0.258·15-s − 0.970·17-s + 0.436·21-s − 0.208·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 0.348·33-s − 0.338·35-s − 1.31·37-s − 0.960·39-s − 0.937·41-s + 0.609·43-s − 0.149·45-s − 3/7·49-s − 0.560·51-s + 0.824·53-s − 0.269·55-s − 1.02·61-s + 0.251·63-s + 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 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
−7.74673326628175605355974976858, −7.20780196542317781931402007860, −6.62982055145948461076398879256, −5.48686143001996106960963032036, −4.66072762268934128349492914666, −4.25054911004514532764637356959, −3.20587612786628764461577555192, −2.34787368795323258730806707505, −1.52848256129043869824842129433, 0,
1.52848256129043869824842129433, 2.34787368795323258730806707505, 3.20587612786628764461577555192, 4.25054911004514532764637356959, 4.66072762268934128349492914666, 5.48686143001996106960963032036, 6.62982055145948461076398879256, 7.20780196542317781931402007860, 7.74673326628175605355974976858