L(s) = 1 | + 3-s − 7-s + 9-s − 11-s + 4·13-s + 6·17-s + 4·19-s − 21-s − 6·23-s + 27-s − 8·29-s − 2·31-s − 33-s + 4·39-s + 2·41-s − 4·43-s + 4·47-s + 49-s + 6·51-s − 14·53-s + 4·57-s + 8·59-s − 10·61-s − 63-s − 4·67-s − 6·69-s − 4·73-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s + 1.10·13-s + 1.45·17-s + 0.917·19-s − 0.218·21-s − 1.25·23-s + 0.192·27-s − 1.48·29-s − 0.359·31-s − 0.174·33-s + 0.640·39-s + 0.312·41-s − 0.609·43-s + 0.583·47-s + 1/7·49-s + 0.840·51-s − 1.92·53-s + 0.529·57-s + 1.04·59-s − 1.28·61-s − 0.125·63-s − 0.488·67-s − 0.722·69-s − 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.830099116\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.830099116\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−12.47837517995636, −12.15833142915855, −11.61097890460817, −11.12988547392936, −10.66419006586864, −10.16562433977106, −9.682064904487949, −9.433190019383484, −8.901127404106027, −8.313550793873824, −7.940273521040869, −7.483858082466225, −7.216949780014204, −6.316984842754249, −6.060899685387585, −5.526266086198006, −5.116794970257896, −4.329096823671785, −3.819000510222552, −3.339554121666554, −3.134231371817340, −2.301756372793908, −1.672558335150573, −1.242247348078039, −0.4196054770582750,
0.4196054770582750, 1.242247348078039, 1.672558335150573, 2.301756372793908, 3.134231371817340, 3.339554121666554, 3.819000510222552, 4.329096823671785, 5.116794970257896, 5.526266086198006, 6.060899685387585, 6.316984842754249, 7.216949780014204, 7.483858082466225, 7.940273521040869, 8.313550793873824, 8.901127404106027, 9.433190019383484, 9.682064904487949, 10.16562433977106, 10.66419006586864, 11.12988547392936, 11.61097890460817, 12.15833142915855, 12.47837517995636