| L(s) = 1 | − 2·7-s + 13-s + 6·17-s − 2·19-s − 5·25-s + 6·29-s − 2·31-s + 2·37-s + 12·41-s + 4·43-s − 3·49-s − 6·53-s + 12·59-s + 2·61-s + 10·67-s + 12·71-s + 14·73-s − 8·79-s + 12·83-s − 2·91-s − 10·97-s − 18·101-s + 16·103-s − 12·107-s + 14·109-s − 6·113-s − 12·119-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 0.277·13-s + 1.45·17-s − 0.458·19-s − 25-s + 1.11·29-s − 0.359·31-s + 0.328·37-s + 1.87·41-s + 0.609·43-s − 3/7·49-s − 0.824·53-s + 1.56·59-s + 0.256·61-s + 1.22·67-s + 1.42·71-s + 1.63·73-s − 0.900·79-s + 1.31·83-s − 0.209·91-s − 1.01·97-s − 1.79·101-s + 1.57·103-s − 1.16·107-s + 1.34·109-s − 0.564·113-s − 1.10·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.589040676\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.589040676\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 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
−9.485148325596294183816572559050, −8.330201821850417900078578930922, −7.76963269770720756784752102354, −6.78013921434981904340142936703, −6.05756828657732859884712721948, −5.31787425835571416225740279689, −4.13529224817546208100485895471, −3.37281563192054858635234459773, −2.33566034645582102228367652701, −0.861350648696305200147674327945,
0.861350648696305200147674327945, 2.33566034645582102228367652701, 3.37281563192054858635234459773, 4.13529224817546208100485895471, 5.31787425835571416225740279689, 6.05756828657732859884712721948, 6.78013921434981904340142936703, 7.76963269770720756784752102354, 8.330201821850417900078578930922, 9.485148325596294183816572559050
