| L(s) = 1 | + 3-s + 9-s − 5·11-s − 2·13-s + 6·17-s − 2·19-s + 5·23-s + 27-s − 5·29-s − 4·31-s − 5·33-s + 37-s − 2·39-s − 12·41-s + 5·43-s + 2·47-s + 6·51-s + 14·53-s − 2·57-s + 2·59-s − 5·67-s + 5·69-s − 9·71-s + 10·73-s + 11·79-s + 81-s + 16·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 1.50·11-s − 0.554·13-s + 1.45·17-s − 0.458·19-s + 1.04·23-s + 0.192·27-s − 0.928·29-s − 0.718·31-s − 0.870·33-s + 0.164·37-s − 0.320·39-s − 1.87·41-s + 0.762·43-s + 0.291·47-s + 0.840·51-s + 1.92·53-s − 0.264·57-s + 0.260·59-s − 0.610·67-s + 0.601·69-s − 1.06·71-s + 1.17·73-s + 1.23·79-s + 1/9·81-s + 1.75·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−15.17793731781928, −14.95443462141668, −14.55443978261040, −13.68220573176656, −13.37410157589181, −12.86965226334995, −12.27200406951012, −11.88124294580780, −10.86533221384382, −10.67053020642405, −9.961277047140201, −9.577435689293223, −8.817013287820100, −8.353809916895013, −7.587248580158297, −7.452091702026214, −6.721852835991608, −5.755570621944582, −5.288545346915580, −4.841960854945648, −3.853701937142902, −3.324077153722310, −2.613666625159299, −2.070606478414251, −1.062619094441708, 0,
1.062619094441708, 2.070606478414251, 2.613666625159299, 3.324077153722310, 3.853701937142902, 4.841960854945648, 5.288545346915580, 5.755570621944582, 6.721852835991608, 7.452091702026214, 7.587248580158297, 8.353809916895013, 8.817013287820100, 9.577435689293223, 9.961277047140201, 10.67053020642405, 10.86533221384382, 11.88124294580780, 12.27200406951012, 12.86965226334995, 13.37410157589181, 13.68220573176656, 14.55443978261040, 14.95443462141668, 15.17793731781928
