| L(s) = 1 | − 3·7-s − 11-s − 13-s − 5·17-s + 8·19-s − 29-s − 3·31-s + 8·37-s + 2·41-s + 8·43-s + 11·47-s + 2·49-s − 11·53-s + 5·59-s + 61-s + 3·67-s + 16·71-s + 4·73-s + 3·77-s − 12·79-s + 3·83-s + 3·91-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 1.13·7-s − 0.301·11-s − 0.277·13-s − 1.21·17-s + 1.83·19-s − 0.185·29-s − 0.538·31-s + 1.31·37-s + 0.312·41-s + 1.21·43-s + 1.60·47-s + 2/7·49-s − 1.51·53-s + 0.650·59-s + 0.128·61-s + 0.366·67-s + 1.89·71-s + 0.468·73-s + 0.341·77-s − 1.35·79-s + 0.329·83-s + 0.314·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.419787309\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.419787309\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.55557084724605, −14.01784929934627, −13.50444216868461, −13.11443484895200, −12.50380794633016, −12.22594087512269, −11.33883984915471, −11.08992229009291, −10.44336531584675, −9.720785123676934, −9.355122299998301, −9.145853598386566, −8.130671854860632, −7.733809486944402, −7.006927107238403, −6.719015947131495, −5.892748024528998, −5.526346694520244, −4.791426614117790, −4.070078705353342, −3.539043005343747, −2.711216604885336, −2.431099096553462, −1.266725974943807, −0.4457739248832207,
0.4457739248832207, 1.266725974943807, 2.431099096553462, 2.711216604885336, 3.539043005343747, 4.070078705353342, 4.791426614117790, 5.526346694520244, 5.892748024528998, 6.719015947131495, 7.006927107238403, 7.733809486944402, 8.130671854860632, 9.145853598386566, 9.355122299998301, 9.720785123676934, 10.44336531584675, 11.08992229009291, 11.33883984915471, 12.22594087512269, 12.50380794633016, 13.11443484895200, 13.50444216868461, 14.01784929934627, 14.55557084724605
