| L(s) = 1 | − 9·3-s − 38·5-s + 49·7-s + 81·9-s − 600·11-s − 674·13-s + 342·15-s + 78·17-s + 916·19-s − 441·21-s + 4.60e3·23-s − 1.68e3·25-s − 729·27-s − 6.81e3·29-s − 7.91e3·31-s + 5.40e3·33-s − 1.86e3·35-s − 9.27e3·37-s + 6.06e3·39-s − 242·41-s − 1.11e3·43-s − 3.07e3·45-s + 2.83e4·47-s + 2.40e3·49-s − 702·51-s + 1.02e4·53-s + 2.28e4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.679·5-s + 0.377·7-s + 1/3·9-s − 1.49·11-s − 1.10·13-s + 0.392·15-s + 0.0654·17-s + 0.582·19-s − 0.218·21-s + 1.81·23-s − 0.537·25-s − 0.192·27-s − 1.50·29-s − 1.47·31-s + 0.863·33-s − 0.256·35-s − 1.11·37-s + 0.638·39-s − 0.0224·41-s − 0.0920·43-s − 0.226·45-s + 1.86·47-s + 1/7·49-s − 0.0377·51-s + 0.500·53-s + 1.01·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.8591142556\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8591142556\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{2} T \) |
| 7 | \( 1 - p^{2} T \) |
| good | 5 | \( 1 + 38 T + p^{5} T^{2} \) |
| 11 | \( 1 + 600 T + p^{5} T^{2} \) |
| 13 | \( 1 + 674 T + p^{5} T^{2} \) |
| 17 | \( 1 - 78 T + p^{5} T^{2} \) |
| 19 | \( 1 - 916 T + p^{5} T^{2} \) |
| 23 | \( 1 - 4604 T + p^{5} T^{2} \) |
| 29 | \( 1 + 6810 T + p^{5} T^{2} \) |
| 31 | \( 1 + 7912 T + p^{5} T^{2} \) |
| 37 | \( 1 + 9274 T + p^{5} T^{2} \) |
| 41 | \( 1 + 242 T + p^{5} T^{2} \) |
| 43 | \( 1 + 1116 T + p^{5} T^{2} \) |
| 47 | \( 1 - 28312 T + p^{5} T^{2} \) |
| 53 | \( 1 - 10230 T + p^{5} T^{2} \) |
| 59 | \( 1 - 4108 T + p^{5} T^{2} \) |
| 61 | \( 1 - 15878 T + p^{5} T^{2} \) |
| 67 | \( 1 - 67668 T + p^{5} T^{2} \) |
| 71 | \( 1 - 67492 T + p^{5} T^{2} \) |
| 73 | \( 1 - 1106 T + p^{5} T^{2} \) |
| 79 | \( 1 + 84152 T + p^{5} T^{2} \) |
| 83 | \( 1 - 2908 T + p^{5} T^{2} \) |
| 89 | \( 1 + 8322 T + p^{5} T^{2} \) |
| 97 | \( 1 - 130810 T + p^{5} T^{2} \) |
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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
−10.89799871667113800280542901114, −9.955881177570997597630359539243, −8.837299870447386077946376398810, −7.48727480343741581831390469333, −7.30705896538217999723570756380, −5.47067545531687311165133028690, −5.01769124645645566366966575590, −3.61021369050209754513287993280, −2.22134449080519971120320051846, −0.50377775677374548265825036963,
0.50377775677374548265825036963, 2.22134449080519971120320051846, 3.61021369050209754513287993280, 5.01769124645645566366966575590, 5.47067545531687311165133028690, 7.30705896538217999723570756380, 7.48727480343741581831390469333, 8.837299870447386077946376398810, 9.955881177570997597630359539243, 10.89799871667113800280542901114
