| L(s) = 1 | + 5·2-s + 9·3-s − 7·4-s + 94·5-s + 45·6-s − 49·7-s − 195·8-s + 81·9-s + 470·10-s + 52·11-s − 63·12-s − 770·13-s − 245·14-s + 846·15-s − 751·16-s − 2.02e3·17-s + 405·18-s + 1.73e3·19-s − 658·20-s − 441·21-s + 260·22-s − 576·23-s − 1.75e3·24-s + 5.71e3·25-s − 3.85e3·26-s + 729·27-s + 343·28-s + ⋯ |
| L(s) = 1 | + 0.883·2-s + 0.577·3-s − 0.218·4-s + 1.68·5-s + 0.510·6-s − 0.377·7-s − 1.07·8-s + 1/3·9-s + 1.48·10-s + 0.129·11-s − 0.126·12-s − 1.26·13-s − 0.334·14-s + 0.970·15-s − 0.733·16-s − 1.69·17-s + 0.294·18-s + 1.10·19-s − 0.367·20-s − 0.218·21-s + 0.114·22-s − 0.227·23-s − 0.621·24-s + 1.82·25-s − 1.11·26-s + 0.192·27-s + 0.0826·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.466125112\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.466125112\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p^{2} T \) |
| 7 | \( 1 + p^{2} T \) |
| good | 2 | \( 1 - 5 T + p^{5} T^{2} \) |
| 5 | \( 1 - 94 T + p^{5} T^{2} \) |
| 11 | \( 1 - 52 T + p^{5} T^{2} \) |
| 13 | \( 1 + 770 T + p^{5} T^{2} \) |
| 17 | \( 1 + 2022 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1732 T + p^{5} T^{2} \) |
| 23 | \( 1 + 576 T + p^{5} T^{2} \) |
| 29 | \( 1 - 5518 T + p^{5} T^{2} \) |
| 31 | \( 1 - 6336 T + p^{5} T^{2} \) |
| 37 | \( 1 + 7338 T + p^{5} T^{2} \) |
| 41 | \( 1 + 3262 T + p^{5} T^{2} \) |
| 43 | \( 1 - 5420 T + p^{5} T^{2} \) |
| 47 | \( 1 - 864 T + p^{5} T^{2} \) |
| 53 | \( 1 - 4182 T + p^{5} T^{2} \) |
| 59 | \( 1 + 11220 T + p^{5} T^{2} \) |
| 61 | \( 1 + 45602 T + p^{5} T^{2} \) |
| 67 | \( 1 - 1396 T + p^{5} T^{2} \) |
| 71 | \( 1 - 18720 T + p^{5} T^{2} \) |
| 73 | \( 1 - 46362 T + p^{5} T^{2} \) |
| 79 | \( 1 - 97424 T + p^{5} T^{2} \) |
| 83 | \( 1 + 81228 T + p^{5} T^{2} \) |
| 89 | \( 1 + 3182 T + p^{5} T^{2} \) |
| 97 | \( 1 - 4914 T + p^{5} T^{2} \) |
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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
−17.32643156645439342246274159127, −15.42310572300675397613978331628, −14.02960867301483434871279482651, −13.60940795219608337703584609636, −12.37687970017926687960974847113, −9.972534427790289468944195469196, −9.050714936104202594141574710225, −6.48631478031342140594096024346, −4.90237485307618850810000414743, −2.60592162683756062197771819964,
2.60592162683756062197771819964, 4.90237485307618850810000414743, 6.48631478031342140594096024346, 9.050714936104202594141574710225, 9.972534427790289468944195469196, 12.37687970017926687960974847113, 13.60940795219608337703584609636, 14.02960867301483434871279482651, 15.42310572300675397613978331628, 17.32643156645439342246274159127
