| L(s) = 1 | − 8·3-s + 14·5-s − 208·7-s − 179·9-s − 536·11-s + 694·13-s − 112·15-s − 1.27e3·17-s + 1.11e3·19-s + 1.66e3·21-s + 3.21e3·23-s − 2.92e3·25-s + 3.37e3·27-s + 2.91e3·29-s − 2.62e3·31-s + 4.28e3·33-s − 2.91e3·35-s − 9.45e3·37-s − 5.55e3·39-s + 170·41-s − 1.99e4·43-s − 2.50e3·45-s + 32·47-s + 2.64e4·49-s + 1.02e4·51-s − 2.21e4·53-s − 7.50e3·55-s + ⋯ |
| L(s) = 1 | − 0.513·3-s + 0.250·5-s − 1.60·7-s − 0.736·9-s − 1.33·11-s + 1.13·13-s − 0.128·15-s − 1.07·17-s + 0.706·19-s + 0.823·21-s + 1.26·23-s − 0.937·25-s + 0.891·27-s + 0.644·29-s − 0.490·31-s + 0.685·33-s − 0.401·35-s − 1.13·37-s − 0.584·39-s + 0.0157·41-s − 1.64·43-s − 0.184·45-s + 0.00211·47-s + 1.57·49-s + 0.550·51-s − 1.08·53-s − 0.334·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 8 T + p^{5} T^{2} \) |
| 5 | \( 1 - 14 T + p^{5} T^{2} \) |
| 7 | \( 1 + 208 T + p^{5} T^{2} \) |
| 11 | \( 1 + 536 T + p^{5} T^{2} \) |
| 13 | \( 1 - 694 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1278 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1112 T + p^{5} T^{2} \) |
| 23 | \( 1 - 3216 T + p^{5} T^{2} \) |
| 29 | \( 1 - 2918 T + p^{5} T^{2} \) |
| 31 | \( 1 + 2624 T + p^{5} T^{2} \) |
| 37 | \( 1 + 9458 T + p^{5} T^{2} \) |
| 41 | \( 1 - 170 T + p^{5} T^{2} \) |
| 43 | \( 1 + 19928 T + p^{5} T^{2} \) |
| 47 | \( 1 - 32 T + p^{5} T^{2} \) |
| 53 | \( 1 + 22178 T + p^{5} T^{2} \) |
| 59 | \( 1 - 41480 T + p^{5} T^{2} \) |
| 61 | \( 1 - 15462 T + p^{5} T^{2} \) |
| 67 | \( 1 + 20744 T + p^{5} T^{2} \) |
| 71 | \( 1 - 28592 T + p^{5} T^{2} \) |
| 73 | \( 1 + 53670 T + p^{5} T^{2} \) |
| 79 | \( 1 + 69152 T + p^{5} T^{2} \) |
| 83 | \( 1 + 37800 T + p^{5} T^{2} \) |
| 89 | \( 1 + 126806 T + p^{5} T^{2} \) |
| 97 | \( 1 - 62290 T + p^{5} T^{2} \) |
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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
−15.58207076215561775169825543226, −13.61336521132877859098282974396, −12.90187517906477616610015363559, −11.31857003666304005985439474032, −10.14003899981572732495857398518, −8.710965820746736160627417431972, −6.70672702287718643748508040768, −5.50818870005420743787505703747, −3.08685227262836591523781351559, 0,
3.08685227262836591523781351559, 5.50818870005420743787505703747, 6.70672702287718643748508040768, 8.710965820746736160627417431972, 10.14003899981572732495857398518, 11.31857003666304005985439474032, 12.90187517906477616610015363559, 13.61336521132877859098282974396, 15.58207076215561775169825543226
