| L(s) = 1 | − 3-s − 2·4-s + 4·7-s − 2·9-s + 11-s + 2·12-s − 2·13-s + 4·16-s + 19-s − 4·21-s − 3·23-s + 5·27-s − 8·28-s − 6·29-s − 7·31-s − 33-s + 4·36-s + 7·37-s + 2·39-s + 10·43-s − 2·44-s − 4·48-s + 9·49-s + 4·52-s − 6·53-s − 57-s + 3·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s + 1.51·7-s − 2/3·9-s + 0.301·11-s + 0.577·12-s − 0.554·13-s + 16-s + 0.229·19-s − 0.872·21-s − 0.625·23-s + 0.962·27-s − 1.51·28-s − 1.11·29-s − 1.25·31-s − 0.174·33-s + 2/3·36-s + 1.15·37-s + 0.320·39-s + 1.52·43-s − 0.301·44-s − 0.577·48-s + 9/7·49-s + 0.554·52-s − 0.824·53-s − 0.132·57-s + 0.390·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.70664303416556724816155022651, −7.50373371660894181179543471243, −6.04933786460783453584413201528, −5.63808381384543581237698198642, −4.86389562366777921547526679517, −4.40334606064562058996569138936, −3.47499273822606942562602848458, −2.23437145276371029093687410514, −1.18804693586949888646681251628, 0,
1.18804693586949888646681251628, 2.23437145276371029093687410514, 3.47499273822606942562602848458, 4.40334606064562058996569138936, 4.86389562366777921547526679517, 5.63808381384543581237698198642, 6.04933786460783453584413201528, 7.50373371660894181179543471243, 7.70664303416556724816155022651
