L(s) = 1 | + 3-s + 5-s + 9-s + 4·13-s + 15-s + 2·19-s − 23-s + 25-s + 27-s + 6·29-s + 2·31-s − 10·37-s + 4·39-s − 6·41-s + 4·43-s + 45-s + 6·47-s − 6·53-s + 2·57-s + 12·59-s + 10·61-s + 4·65-s + 4·67-s − 69-s − 14·73-s + 75-s − 8·79-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s + 1.10·13-s + 0.258·15-s + 0.458·19-s − 0.208·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.359·31-s − 1.64·37-s + 0.640·39-s − 0.937·41-s + 0.609·43-s + 0.149·45-s + 0.875·47-s − 0.824·53-s + 0.264·57-s + 1.56·59-s + 1.28·61-s + 0.496·65-s + 0.488·67-s − 0.120·69-s − 1.63·73-s + 0.115·75-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−13.08146761615733, −12.63199263093202, −12.11044736263909, −11.64886915499620, −11.19701147583091, −10.52074309315884, −10.27033060402042, −9.824745245296897, −9.261214361265809, −8.717231398823558, −8.461691598725053, −8.069168151939458, −7.284842886791161, −6.952995754744127, −6.441607049268174, −5.898629270942905, −5.411562865519125, −4.882452853160453, −4.279638981522740, −3.643346475323304, −3.360597115054769, −2.587016489630880, −2.207119327188814, −1.323396923946691, −1.095827890429017, 0,
1.095827890429017, 1.323396923946691, 2.207119327188814, 2.587016489630880, 3.360597115054769, 3.643346475323304, 4.279638981522740, 4.882452853160453, 5.411562865519125, 5.898629270942905, 6.441607049268174, 6.952995754744127, 7.284842886791161, 8.069168151939458, 8.461691598725053, 8.717231398823558, 9.261214361265809, 9.824745245296897, 10.27033060402042, 10.52074309315884, 11.19701147583091, 11.64886915499620, 12.11044736263909, 12.63199263093202, 13.08146761615733