L(s) = 1 | + 5-s + 2·7-s + 2·11-s + 2·13-s + 4·17-s − 6·19-s + 6·23-s + 25-s − 10·29-s + 8·31-s + 2·35-s − 2·37-s − 2·41-s − 43-s − 2·47-s − 3·49-s + 10·53-s + 2·55-s − 2·59-s + 12·61-s + 2·65-s − 12·67-s + 16·71-s + 16·73-s + 4·77-s + 8·79-s + 12·83-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.755·7-s + 0.603·11-s + 0.554·13-s + 0.970·17-s − 1.37·19-s + 1.25·23-s + 1/5·25-s − 1.85·29-s + 1.43·31-s + 0.338·35-s − 0.328·37-s − 0.312·41-s − 0.152·43-s − 0.291·47-s − 3/7·49-s + 1.37·53-s + 0.269·55-s − 0.260·59-s + 1.53·61-s + 0.248·65-s − 1.46·67-s + 1.89·71-s + 1.87·73-s + 0.455·77-s + 0.900·79-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.980697366\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.980697366\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 43 | \( 1 + T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.45994372226780, −13.22098521728065, −12.46516952998044, −12.21415668220293, −11.43387735921668, −11.19730187670142, −10.66229273682474, −10.20661931232345, −9.567120093892552, −9.156108032395344, −8.629884112647921, −8.121627889363340, −7.768672741713220, −6.888330918046123, −6.648373646987520, −6.041768405365033, −5.352631186924323, −5.091528746299688, −4.340766226704009, −3.776463869790034, −3.298965147930299, −2.417406602415846, −1.905401834848984, −1.263110041300922, −0.6451280424215831,
0.6451280424215831, 1.263110041300922, 1.905401834848984, 2.417406602415846, 3.298965147930299, 3.776463869790034, 4.340766226704009, 5.091528746299688, 5.352631186924323, 6.041768405365033, 6.648373646987520, 6.888330918046123, 7.768672741713220, 8.121627889363340, 8.629884112647921, 9.156108032395344, 9.567120093892552, 10.20661931232345, 10.66229273682474, 11.19730187670142, 11.43387735921668, 12.21415668220293, 12.46516952998044, 13.22098521728065, 13.45994372226780