| L(s) = 1 | − 2-s + 4-s + 4·7-s − 8-s − 6·11-s − 4·14-s + 16-s − 4·17-s − 19-s + 6·22-s + 4·23-s + 4·28-s − 10·29-s − 2·31-s − 32-s + 4·34-s + 4·37-s + 38-s + 10·41-s + 12·43-s − 6·44-s − 4·46-s + 9·49-s + 6·53-s − 4·56-s + 10·58-s + 4·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.51·7-s − 0.353·8-s − 1.80·11-s − 1.06·14-s + 1/4·16-s − 0.970·17-s − 0.229·19-s + 1.27·22-s + 0.834·23-s + 0.755·28-s − 1.85·29-s − 0.359·31-s − 0.176·32-s + 0.685·34-s + 0.657·37-s + 0.162·38-s + 1.56·41-s + 1.82·43-s − 0.904·44-s − 0.589·46-s + 9/7·49-s + 0.824·53-s − 0.534·56-s + 1.31·58-s + 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−7.64059398920883183641947574385, −7.13125467740691937671729094497, −5.93741642032031697278124326609, −5.42582262254417695045791543671, −4.71465605744879262461657899692, −3.97364775306697559042605615032, −2.60802948090383273512278853257, −2.26494797074762937220699479188, −1.21175935536621145873262555618, 0,
1.21175935536621145873262555618, 2.26494797074762937220699479188, 2.60802948090383273512278853257, 3.97364775306697559042605615032, 4.71465605744879262461657899692, 5.42582262254417695045791543671, 5.93741642032031697278124326609, 7.13125467740691937671729094497, 7.64059398920883183641947574385
