L(s) = 1 | + 2·5-s + 4·7-s − 2·11-s + 7·13-s + 4·17-s + 6·19-s − 23-s − 25-s − 5·29-s − 3·31-s + 8·35-s + 2·37-s + 9·41-s − 8·43-s − 47-s + 9·49-s + 6·53-s − 4·55-s − 8·59-s − 10·61-s + 14·65-s − 2·67-s − 13·71-s − 3·73-s − 8·77-s − 6·79-s + 8·85-s + ⋯ |
L(s) = 1 | + 0.894·5-s + 1.51·7-s − 0.603·11-s + 1.94·13-s + 0.970·17-s + 1.37·19-s − 0.208·23-s − 1/5·25-s − 0.928·29-s − 0.538·31-s + 1.35·35-s + 0.328·37-s + 1.40·41-s − 1.21·43-s − 0.145·47-s + 9/7·49-s + 0.824·53-s − 0.539·55-s − 1.04·59-s − 1.28·61-s + 1.73·65-s − 0.244·67-s − 1.54·71-s − 0.351·73-s − 0.911·77-s − 0.675·79-s + 0.867·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.073998827\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.073998827\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−8.615226630901928818898861906276, −7.82822497403279594511533386356, −7.42305646608664361372230668147, −6.01856800181411305648298215512, −5.70766718008796804190096926738, −4.98146621221420200689209256046, −3.95108373398539735969662581831, −3.02072303977103941908898175729, −1.75431467900160514570712646468, −1.23359092033143054304220976612,
1.23359092033143054304220976612, 1.75431467900160514570712646468, 3.02072303977103941908898175729, 3.95108373398539735969662581831, 4.98146621221420200689209256046, 5.70766718008796804190096926738, 6.01856800181411305648298215512, 7.42305646608664361372230668147, 7.82822497403279594511533386356, 8.615226630901928818898861906276