| L(s) = 1 | − 2·4-s + 5-s + 3·7-s + 5·11-s + 4·16-s + 7·17-s − 2·20-s + 4·23-s − 4·25-s − 6·28-s + 3·35-s − 43-s − 10·44-s − 13·47-s + 2·49-s + 5·55-s + 15·61-s − 8·64-s − 14·68-s − 11·73-s + 15·77-s + 4·80-s + 16·83-s + 7·85-s − 8·92-s + 8·100-s − 10·101-s + ⋯ |
| L(s) = 1 | − 4-s + 0.447·5-s + 1.13·7-s + 1.50·11-s + 16-s + 1.69·17-s − 0.447·20-s + 0.834·23-s − 4/5·25-s − 1.13·28-s + 0.507·35-s − 0.152·43-s − 1.50·44-s − 1.89·47-s + 2/7·49-s + 0.674·55-s + 1.92·61-s − 64-s − 1.69·68-s − 1.28·73-s + 1.70·77-s + 0.447·80-s + 1.75·83-s + 0.759·85-s − 0.834·92-s + 4/5·100-s − 0.995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.220207638\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.220207638\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 15 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−8.580042248187093986713935126037, −8.107116855348831176369596581034, −7.27943314894574780769453240025, −6.27245985312872964632310218742, −5.45289188915211295785069372208, −4.88187668285910676494600805335, −4.00355565153622805697617293535, −3.28074247116569086926038001604, −1.71787306481732958102463482448, −1.01885130548047428295164993318,
1.01885130548047428295164993318, 1.71787306481732958102463482448, 3.28074247116569086926038001604, 4.00355565153622805697617293535, 4.88187668285910676494600805335, 5.45289188915211295785069372208, 6.27245985312872964632310218742, 7.27943314894574780769453240025, 8.107116855348831176369596581034, 8.580042248187093986713935126037
