| L(s) = 1 | + 3-s + 2·7-s + 9-s − 11-s − 2·13-s − 4·17-s + 6·19-s + 2·21-s + 27-s + 8·29-s − 8·31-s − 33-s + 10·37-s − 2·39-s + 8·41-s − 2·43-s + 8·47-s − 3·49-s − 4·51-s − 2·53-s + 6·57-s − 12·59-s − 10·61-s + 2·63-s + 12·67-s + 8·71-s − 6·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.755·7-s + 1/3·9-s − 0.301·11-s − 0.554·13-s − 0.970·17-s + 1.37·19-s + 0.436·21-s + 0.192·27-s + 1.48·29-s − 1.43·31-s − 0.174·33-s + 1.64·37-s − 0.320·39-s + 1.24·41-s − 0.304·43-s + 1.16·47-s − 3/7·49-s − 0.560·51-s − 0.274·53-s + 0.794·57-s − 1.56·59-s − 1.28·61-s + 0.251·63-s + 1.46·67-s + 0.949·71-s − 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.360020141\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.360020141\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−14.31935275303326, −14.01606999299161, −13.63652555548861, −12.84021155528156, −12.56571499194393, −11.91006413629584, −11.26946424219854, −10.98053828852177, −10.32897594812840, −9.680272689482370, −9.238621502737543, −8.820969006991665, −8.011128694537708, −7.723339213185481, −7.273790978831981, −6.541578117398887, −5.913207637037878, −5.189617420822081, −4.676478227732120, −4.230081658132749, −3.377780527310091, −2.732929714545610, −2.221511333641911, −1.429665568200662, −0.6297637240791607,
0.6297637240791607, 1.429665568200662, 2.221511333641911, 2.732929714545610, 3.377780527310091, 4.230081658132749, 4.676478227732120, 5.189617420822081, 5.913207637037878, 6.541578117398887, 7.273790978831981, 7.723339213185481, 8.011128694537708, 8.820969006991665, 9.238621502737543, 9.680272689482370, 10.32897594812840, 10.98053828852177, 11.26946424219854, 11.91006413629584, 12.56571499194393, 12.84021155528156, 13.63652555548861, 14.01606999299161, 14.31935275303326
