| L(s) = 1 | + 3-s − 2·4-s + 5-s + 9-s − 11-s − 2·12-s − 13-s + 15-s + 4·16-s + 4·17-s − 2·19-s − 2·20-s + 23-s + 25-s + 27-s − 9·29-s − 33-s − 2·36-s − 39-s − 5·41-s − 11·43-s + 2·44-s + 45-s − 9·47-s + 4·48-s + 4·51-s + 2·52-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 0.447·5-s + 1/3·9-s − 0.301·11-s − 0.577·12-s − 0.277·13-s + 0.258·15-s + 16-s + 0.970·17-s − 0.458·19-s − 0.447·20-s + 0.208·23-s + 1/5·25-s + 0.192·27-s − 1.67·29-s − 0.174·33-s − 1/3·36-s − 0.160·39-s − 0.780·41-s − 1.67·43-s + 0.301·44-s + 0.149·45-s − 1.31·47-s + 0.577·48-s + 0.560·51-s + 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.62740807273867899191391773229, −6.91765721455926191124643774497, −5.94792050633979382510521231316, −5.28040303953114681086446027201, −4.74627084586935276189568761358, −3.75396014378521258503865061098, −3.28293503395881979047995072436, −2.21031462860355403391568753091, −1.31177757297713829137286502882, 0,
1.31177757297713829137286502882, 2.21031462860355403391568753091, 3.28293503395881979047995072436, 3.75396014378521258503865061098, 4.74627084586935276189568761358, 5.28040303953114681086446027201, 5.94792050633979382510521231316, 6.91765721455926191124643774497, 7.62740807273867899191391773229
