| L(s) = 1 | + 3-s − 2·5-s + 7-s + 9-s + 6·11-s − 4·13-s − 2·15-s − 17-s + 2·19-s + 21-s − 8·23-s − 25-s + 27-s + 8·31-s + 6·33-s − 2·35-s + 4·37-s − 4·39-s + 2·41-s − 12·43-s − 2·45-s + 8·47-s + 49-s − 51-s + 6·53-s − 12·55-s + 2·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s + 1.80·11-s − 1.10·13-s − 0.516·15-s − 0.242·17-s + 0.458·19-s + 0.218·21-s − 1.66·23-s − 1/5·25-s + 0.192·27-s + 1.43·31-s + 1.04·33-s − 0.338·35-s + 0.657·37-s − 0.640·39-s + 0.312·41-s − 1.82·43-s − 0.298·45-s + 1.16·47-s + 1/7·49-s − 0.140·51-s + 0.824·53-s − 1.61·55-s + 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.373963738\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.373963738\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−15.32354944250208, −15.01092465881567, −14.42513049983657, −13.81748929864501, −13.71370888778603, −12.55413791848940, −12.08002252448970, −11.80566240525972, −11.37688569172461, −10.47210773818745, −9.794887073208231, −9.498511790449020, −8.710078950973265, −8.244141292976892, −7.650710541563905, −7.208255349040379, −6.470978074837679, −5.935384074291418, −4.856997039189566, −4.334035252881483, −3.891244220954561, −3.201439796650788, −2.301798230852890, −1.595510573936657, −0.6104157445440720,
0.6104157445440720, 1.595510573936657, 2.301798230852890, 3.201439796650788, 3.891244220954561, 4.334035252881483, 4.856997039189566, 5.935384074291418, 6.470978074837679, 7.208255349040379, 7.650710541563905, 8.244141292976892, 8.710078950973265, 9.498511790449020, 9.794887073208231, 10.47210773818745, 11.37688569172461, 11.80566240525972, 12.08002252448970, 12.55413791848940, 13.71370888778603, 13.81748929864501, 14.42513049983657, 15.01092465881567, 15.32354944250208
