| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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
−13.44510751196445172563506890025, −11.87941328228924390624057934249, −11.33835919481616859473873863468, −10.13688581848830139880630464405, −8.375312802600282497139234011824, −7.54035521012916284323275145076, −5.60710375224466644935115887144, −4.64804404780004490838770542812, −2.28886993127383475540982002662, 0,
2.28886993127383475540982002662, 4.64804404780004490838770542812, 5.60710375224466644935115887144, 7.54035521012916284323275145076, 8.375312802600282497139234011824, 10.13688581848830139880630464405, 11.33835919481616859473873863468, 11.87941328228924390624057934249, 13.44510751196445172563506890025
