| L(s) = 1 | + 2-s + 4-s + 4·5-s + 7-s + 8-s − 3·9-s + 4·10-s + 2·13-s + 14-s + 16-s + 6·17-s − 3·18-s − 4·19-s + 4·20-s + 11·25-s + 2·26-s + 28-s − 2·29-s + 2·31-s + 32-s + 6·34-s + 4·35-s − 3·36-s − 2·37-s − 4·38-s + 4·40-s − 2·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.78·5-s + 0.377·7-s + 0.353·8-s − 9-s + 1.26·10-s + 0.554·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.707·18-s − 0.917·19-s + 0.894·20-s + 11/5·25-s + 0.392·26-s + 0.188·28-s − 0.371·29-s + 0.359·31-s + 0.176·32-s + 1.02·34-s + 0.676·35-s − 1/2·36-s − 0.328·37-s − 0.648·38-s + 0.632·40-s − 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25886 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25886 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.067201582\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.067201582\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 43 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 18 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−14.93213760835141, −14.70779141364656, −14.12456270019432, −13.87312821326539, −13.21812991388925, −12.87524251047180, −12.11637998024009, −11.68602646023554, −10.85525664700532, −10.57144565345334, −9.957283227381822, −9.327439100888339, −8.784571428124875, −8.147386635670875, −7.534644100556365, −6.518516842338310, −6.207103568993711, −5.748162996917822, −5.129673207739386, −4.727528421275268, −3.535899874690122, −3.124795737381432, −2.207121280365740, −1.802625373872047, −0.8732888741475411,
0.8732888741475411, 1.802625373872047, 2.207121280365740, 3.124795737381432, 3.535899874690122, 4.727528421275268, 5.129673207739386, 5.748162996917822, 6.207103568993711, 6.518516842338310, 7.534644100556365, 8.147386635670875, 8.784571428124875, 9.327439100888339, 9.957283227381822, 10.57144565345334, 10.85525664700532, 11.68602646023554, 12.11637998024009, 12.87524251047180, 13.21812991388925, 13.87312821326539, 14.12456270019432, 14.70779141364656, 14.93213760835141
