| L(s) = 1 | + 1.95·2-s + 1.82·4-s + 3.56·7-s − 0.340·8-s − 5.56·11-s − 5.26·13-s + 6.96·14-s − 4.31·16-s + 1.40·17-s + 19-s − 10.8·22-s − 6.96·23-s − 10.3·26-s + 6.50·28-s − 1.40·29-s + 1.75·31-s − 7.76·32-s + 2.75·34-s − 3.61·37-s + 1.95·38-s − 4.34·41-s + 3.56·43-s − 10.1·44-s − 13.6·46-s − 8.26·47-s + 5.69·49-s − 9.61·52-s + ⋯ |
| L(s) = 1 | + 1.38·2-s + 0.912·4-s + 1.34·7-s − 0.120·8-s − 1.67·11-s − 1.46·13-s + 1.86·14-s − 1.07·16-s + 0.341·17-s + 0.229·19-s − 2.31·22-s − 1.45·23-s − 2.02·26-s + 1.22·28-s − 0.261·29-s + 0.315·31-s − 1.37·32-s + 0.471·34-s − 0.594·37-s + 0.317·38-s − 0.679·41-s + 0.543·43-s − 1.53·44-s − 2.00·46-s − 1.20·47-s + 0.813·49-s − 1.33·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 1.95T + 2T^{2} \) |
| 7 | \( 1 - 3.56T + 7T^{2} \) |
| 11 | \( 1 + 5.56T + 11T^{2} \) |
| 13 | \( 1 + 5.26T + 13T^{2} \) |
| 17 | \( 1 - 1.40T + 17T^{2} \) |
| 23 | \( 1 + 6.96T + 23T^{2} \) |
| 29 | \( 1 + 1.40T + 29T^{2} \) |
| 31 | \( 1 - 1.75T + 31T^{2} \) |
| 37 | \( 1 + 3.61T + 37T^{2} \) |
| 41 | \( 1 + 4.34T + 41T^{2} \) |
| 43 | \( 1 - 3.56T + 43T^{2} \) |
| 47 | \( 1 + 8.26T + 47T^{2} \) |
| 53 | \( 1 + 7.61T + 53T^{2} \) |
| 59 | \( 1 + 9.47T + 59T^{2} \) |
| 61 | \( 1 - 9.21T + 61T^{2} \) |
| 67 | \( 1 - 4.76T + 67T^{2} \) |
| 71 | \( 1 - 14.0T + 71T^{2} \) |
| 73 | \( 1 + 6.59T + 73T^{2} \) |
| 79 | \( 1 - 5.47T + 79T^{2} \) |
| 83 | \( 1 + 4.15T + 83T^{2} \) |
| 89 | \( 1 - 9.23T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−7.86889999139568826536470957187, −7.34938715083010108415824573510, −6.29819399791246483203111757531, −5.37278531923744420303442245109, −5.04777032744290377370965783458, −4.52587171547591653368613668987, −3.49063016580615184669288296634, −2.55223325632033865107478474355, −1.93431037323643681256023393008, 0,
1.93431037323643681256023393008, 2.55223325632033865107478474355, 3.49063016580615184669288296634, 4.52587171547591653368613668987, 5.04777032744290377370965783458, 5.37278531923744420303442245109, 6.29819399791246483203111757531, 7.34938715083010108415824573510, 7.86889999139568826536470957187
