| L(s) = 1 | + 0.0643·3-s − 2.31·5-s + 1.63·7-s − 2.99·9-s + 1.15·13-s − 0.148·15-s + 3.36·17-s + 5.83·19-s + 0.104·21-s − 7.86·23-s + 0.350·25-s − 0.385·27-s + 5.92·29-s − 3.61·31-s − 3.77·35-s + 0.362·37-s + 0.0743·39-s − 4.40·41-s − 5.15·43-s + 6.92·45-s − 1.15·47-s − 4.34·49-s + 0.216·51-s − 6.14·53-s + 0.375·57-s + 8.12·59-s + 8.82·61-s + ⋯ |
| L(s) = 1 | + 0.0371·3-s − 1.03·5-s + 0.616·7-s − 0.998·9-s + 0.320·13-s − 0.0384·15-s + 0.815·17-s + 1.33·19-s + 0.0228·21-s − 1.63·23-s + 0.0700·25-s − 0.0742·27-s + 1.10·29-s − 0.648·31-s − 0.637·35-s + 0.0595·37-s + 0.0119·39-s − 0.688·41-s − 0.786·43-s + 1.03·45-s − 0.168·47-s − 0.620·49-s + 0.0302·51-s − 0.843·53-s + 0.0497·57-s + 1.05·59-s + 1.13·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 0.0643T + 3T^{2} \) |
| 5 | \( 1 + 2.31T + 5T^{2} \) |
| 7 | \( 1 - 1.63T + 7T^{2} \) |
| 13 | \( 1 - 1.15T + 13T^{2} \) |
| 17 | \( 1 - 3.36T + 17T^{2} \) |
| 19 | \( 1 - 5.83T + 19T^{2} \) |
| 23 | \( 1 + 7.86T + 23T^{2} \) |
| 29 | \( 1 - 5.92T + 29T^{2} \) |
| 31 | \( 1 + 3.61T + 31T^{2} \) |
| 37 | \( 1 - 0.362T + 37T^{2} \) |
| 41 | \( 1 + 4.40T + 41T^{2} \) |
| 43 | \( 1 + 5.15T + 43T^{2} \) |
| 47 | \( 1 + 1.15T + 47T^{2} \) |
| 53 | \( 1 + 6.14T + 53T^{2} \) |
| 59 | \( 1 - 8.12T + 59T^{2} \) |
| 61 | \( 1 - 8.82T + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 + 15.0T + 71T^{2} \) |
| 73 | \( 1 - 13.0T + 73T^{2} \) |
| 79 | \( 1 - 2.07T + 79T^{2} \) |
| 83 | \( 1 + 8.94T + 83T^{2} \) |
| 89 | \( 1 - 13.9T + 89T^{2} \) |
| 97 | \( 1 - 0.00798T + 97T^{2} \) |
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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
−7.73030519142363399042660603829, −6.94221961929963925995740483810, −6.04478043605088898077845986480, −5.37591723460893189669061890469, −4.74336263606720155163104282139, −3.69762718790402184484569366385, −3.37180485772636108063604241743, −2.28524569727403331293985629422, −1.16716977753463265865640322435, 0,
1.16716977753463265865640322435, 2.28524569727403331293985629422, 3.37180485772636108063604241743, 3.69762718790402184484569366385, 4.74336263606720155163104282139, 5.37591723460893189669061890469, 6.04478043605088898077845986480, 6.94221961929963925995740483810, 7.73030519142363399042660603829
