| L(s) = 1 | + 3·3-s + 16·5-s − 34·7-s + 9·9-s + 48·11-s + 58·13-s + 48·15-s − 17·17-s − 20·19-s − 102·21-s − 58·23-s + 131·25-s + 27·27-s + 218·31-s + 144·33-s − 544·35-s + 184·37-s + 174·39-s − 138·41-s − 148·43-s + 144·45-s + 516·47-s + 813·49-s − 51·51-s − 162·53-s + 768·55-s − 60·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.43·5-s − 1.83·7-s + 1/3·9-s + 1.31·11-s + 1.23·13-s + 0.826·15-s − 0.242·17-s − 0.241·19-s − 1.05·21-s − 0.525·23-s + 1.04·25-s + 0.192·27-s + 1.26·31-s + 0.759·33-s − 2.62·35-s + 0.817·37-s + 0.714·39-s − 0.525·41-s − 0.524·43-s + 0.477·45-s + 1.60·47-s + 2.37·49-s − 0.140·51-s − 0.419·53-s + 1.88·55-s − 0.139·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.181079685\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.181079685\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 17 | \( 1 + p T \) |
| good | 5 | \( 1 - 16 T + p^{3} T^{2} \) |
| 7 | \( 1 + 34 T + p^{3} T^{2} \) |
| 11 | \( 1 - 48 T + p^{3} T^{2} \) |
| 13 | \( 1 - 58 T + p^{3} T^{2} \) |
| 19 | \( 1 + 20 T + p^{3} T^{2} \) |
| 23 | \( 1 + 58 T + p^{3} T^{2} \) |
| 29 | \( 1 + p^{3} T^{2} \) |
| 31 | \( 1 - 218 T + p^{3} T^{2} \) |
| 37 | \( 1 - 184 T + p^{3} T^{2} \) |
| 41 | \( 1 + 138 T + p^{3} T^{2} \) |
| 43 | \( 1 + 148 T + p^{3} T^{2} \) |
| 47 | \( 1 - 516 T + p^{3} T^{2} \) |
| 53 | \( 1 + 162 T + p^{3} T^{2} \) |
| 59 | \( 1 - 180 T + p^{3} T^{2} \) |
| 61 | \( 1 - 152 T + p^{3} T^{2} \) |
| 67 | \( 1 - 956 T + p^{3} T^{2} \) |
| 71 | \( 1 - 538 T + p^{3} T^{2} \) |
| 73 | \( 1 + 462 T + p^{3} T^{2} \) |
| 79 | \( 1 + 390 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1268 T + p^{3} T^{2} \) |
| 89 | \( 1 + 770 T + p^{3} T^{2} \) |
| 97 | \( 1 - 494 T + p^{3} T^{2} \) |
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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
−9.811779945015893893702090941985, −9.122368328359962532073052796973, −8.517507013343752752927762241117, −6.91581136453736567167375056743, −6.32736450995507122001547863495, −5.85130466298833608001840679402, −4.14740008167021153923327752951, −3.29667745622548910715738108794, −2.24838430323256907210298650624, −1.01538303668818242991611084542,
1.01538303668818242991611084542, 2.24838430323256907210298650624, 3.29667745622548910715738108794, 4.14740008167021153923327752951, 5.85130466298833608001840679402, 6.32736450995507122001547863495, 6.91581136453736567167375056743, 8.517507013343752752927762241117, 9.122368328359962532073052796973, 9.811779945015893893702090941985
