| L(s) = 1 | − 5-s − 4·7-s + 11-s − 4·13-s − 4·19-s + 6·23-s + 25-s + 6·29-s + 8·31-s + 4·35-s + 2·37-s − 6·41-s + 8·43-s − 6·47-s + 9·49-s + 6·53-s − 55-s + 12·59-s + 2·61-s + 4·65-s − 10·67-s + 12·71-s − 16·73-s − 4·77-s + 8·79-s − 6·89-s + 16·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.51·7-s + 0.301·11-s − 1.10·13-s − 0.917·19-s + 1.25·23-s + 1/5·25-s + 1.11·29-s + 1.43·31-s + 0.676·35-s + 0.328·37-s − 0.937·41-s + 1.21·43-s − 0.875·47-s + 9/7·49-s + 0.824·53-s − 0.134·55-s + 1.56·59-s + 0.256·61-s + 0.496·65-s − 1.22·67-s + 1.42·71-s − 1.87·73-s − 0.455·77-s + 0.900·79-s − 0.635·89-s + 1.67·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.063789879\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.063789879\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 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 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−9.143388368464316920898426683170, −8.521834734599108865403120172911, −7.47601643346092433487754151539, −6.73777549197838954142221486732, −6.25760067471114060229971558685, −5.03835627849849618692157113012, −4.22887800367054726811563143272, −3.20327356214192940251353293177, −2.49581293620386904412683138099, −0.66380362494939169440974615231,
0.66380362494939169440974615231, 2.49581293620386904412683138099, 3.20327356214192940251353293177, 4.22887800367054726811563143272, 5.03835627849849618692157113012, 6.25760067471114060229971558685, 6.73777549197838954142221486732, 7.47601643346092433487754151539, 8.521834734599108865403120172911, 9.143388368464316920898426683170
