| L(s) = 1 | − 2-s + 4-s − 8-s + 2·11-s − 4·13-s + 16-s + 5·17-s + 4·19-s − 2·22-s + 5·23-s + 4·26-s + 6·29-s + 11·31-s − 32-s − 5·34-s − 8·37-s − 4·38-s + 5·41-s + 2·44-s − 5·46-s + 47-s − 4·52-s + 12·53-s − 6·58-s − 2·59-s − 10·61-s − 11·62-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 0.603·11-s − 1.10·13-s + 1/4·16-s + 1.21·17-s + 0.917·19-s − 0.426·22-s + 1.04·23-s + 0.784·26-s + 1.11·29-s + 1.97·31-s − 0.176·32-s − 0.857·34-s − 1.31·37-s − 0.648·38-s + 0.780·41-s + 0.301·44-s − 0.737·46-s + 0.145·47-s − 0.554·52-s + 1.64·53-s − 0.787·58-s − 0.260·59-s − 1.28·61-s − 1.39·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.973794342\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.973794342\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 11 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−15.60643836844839, −15.05279208108060, −14.50976615395759, −13.95075806169164, −13.53932920788297, −12.51746067155406, −12.11267088756181, −11.86060135834894, −11.13133741923603, −10.30603767127920, −10.07116329007853, −9.483715926180864, −8.864949474932961, −8.351078936731336, −7.516756752038668, −7.310331541318576, −6.531091378857055, −5.950073149848846, −5.081025211297374, −4.692678442274727, −3.609792594722878, −2.985955415379053, −2.354026898733695, −1.243179496386653, −0.7353338229709871,
0.7353338229709871, 1.243179496386653, 2.354026898733695, 2.985955415379053, 3.609792594722878, 4.692678442274727, 5.081025211297374, 5.950073149848846, 6.531091378857055, 7.310331541318576, 7.516756752038668, 8.351078936731336, 8.864949474932961, 9.483715926180864, 10.07116329007853, 10.30603767127920, 11.13133741923603, 11.86060135834894, 12.11267088756181, 12.51746067155406, 13.53932920788297, 13.95075806169164, 14.50976615395759, 15.05279208108060, 15.60643836844839
