L(s) = 1 | + 2-s − 4-s + 7-s − 3·8-s − 2·13-s + 14-s − 16-s − 7·17-s + 4·19-s + 6·23-s − 2·26-s − 28-s − 2·29-s − 11·31-s + 5·32-s − 7·34-s + 3·37-s + 4·38-s − 6·41-s + 43-s + 6·46-s − 3·47-s + 49-s + 2·52-s − 9·53-s − 3·56-s − 2·58-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.377·7-s − 1.06·8-s − 0.554·13-s + 0.267·14-s − 1/4·16-s − 1.69·17-s + 0.917·19-s + 1.25·23-s − 0.392·26-s − 0.188·28-s − 0.371·29-s − 1.97·31-s + 0.883·32-s − 1.20·34-s + 0.493·37-s + 0.648·38-s − 0.937·41-s + 0.152·43-s + 0.884·46-s − 0.437·47-s + 1/7·49-s + 0.277·52-s − 1.23·53-s − 0.400·56-s − 0.262·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5035510002\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5035510002\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 11 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−13.10379092576146, −12.77076046011391, −12.35206563021852, −11.61606246160697, −11.39961464680965, −10.89037098505452, −10.36261534675986, −9.640194484640466, −9.270003813706730, −8.871871334465273, −8.549248244604668, −7.619239970339979, −7.439802152319591, −6.759447540227484, −6.245183565434154, −5.661205572786629, −5.079117433820395, −4.839660907682003, −4.296507943545747, −3.727172161920382, −3.104323206437876, −2.681762316229821, −1.862732903028085, −1.276264385642762, −0.1786412922407533,
0.1786412922407533, 1.276264385642762, 1.862732903028085, 2.681762316229821, 3.104323206437876, 3.727172161920382, 4.296507943545747, 4.839660907682003, 5.079117433820395, 5.661205572786629, 6.245183565434154, 6.759447540227484, 7.439802152319591, 7.619239970339979, 8.549248244604668, 8.871871334465273, 9.270003813706730, 9.640194484640466, 10.36261534675986, 10.89037098505452, 11.39961464680965, 11.61606246160697, 12.35206563021852, 12.77076046011391, 13.10379092576146