| L(s) = 1 | − 6·2-s − 9·3-s + 4·4-s + 78·5-s + 54·6-s + 49·7-s + 168·8-s + 81·9-s − 468·10-s + 444·11-s − 36·12-s − 442·13-s − 294·14-s − 702·15-s − 1.13e3·16-s − 126·17-s − 486·18-s + 2.68e3·19-s + 312·20-s − 441·21-s − 2.66e3·22-s + 4.20e3·23-s − 1.51e3·24-s + 2.95e3·25-s + 2.65e3·26-s − 729·27-s + 196·28-s + ⋯ |
| L(s) = 1 | − 1.06·2-s − 0.577·3-s + 1/8·4-s + 1.39·5-s + 0.612·6-s + 0.377·7-s + 0.928·8-s + 1/3·9-s − 1.47·10-s + 1.10·11-s − 0.0721·12-s − 0.725·13-s − 0.400·14-s − 0.805·15-s − 1.10·16-s − 0.105·17-s − 0.353·18-s + 1.70·19-s + 0.174·20-s − 0.218·21-s − 1.17·22-s + 1.65·23-s − 0.535·24-s + 0.946·25-s + 0.769·26-s − 0.192·27-s + 0.0472·28-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
| \(L(3)\) |
\(\approx\) |
\(0.8679490723\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8679490723\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 3 p T + p^{5} T^{2} \) |
| 5 | \( 1 - 78 T + p^{5} T^{2} \) |
| 11 | \( 1 - 444 T + p^{5} T^{2} \) |
| 13 | \( 1 + 34 p T + p^{5} T^{2} \) |
| 17 | \( 1 + 126 T + p^{5} T^{2} \) |
| 19 | \( 1 - 2684 T + p^{5} T^{2} \) |
| 23 | \( 1 - 4200 T + p^{5} T^{2} \) |
| 29 | \( 1 + 5442 T + p^{5} T^{2} \) |
| 31 | \( 1 - 80 T + p^{5} T^{2} \) |
| 37 | \( 1 + 5434 T + p^{5} T^{2} \) |
| 41 | \( 1 - 7962 T + p^{5} T^{2} \) |
| 43 | \( 1 + 268 p T + p^{5} T^{2} \) |
| 47 | \( 1 + 13920 T + p^{5} T^{2} \) |
| 53 | \( 1 + 9594 T + p^{5} T^{2} \) |
| 59 | \( 1 - 27492 T + p^{5} T^{2} \) |
| 61 | \( 1 - 49478 T + p^{5} T^{2} \) |
| 67 | \( 1 + 59356 T + p^{5} T^{2} \) |
| 71 | \( 1 - 32040 T + p^{5} T^{2} \) |
| 73 | \( 1 + 61846 T + p^{5} T^{2} \) |
| 79 | \( 1 + 65776 T + p^{5} T^{2} \) |
| 83 | \( 1 - 40188 T + p^{5} T^{2} \) |
| 89 | \( 1 + 7974 T + p^{5} T^{2} \) |
| 97 | \( 1 + 143662 T + p^{5} T^{2} \) |
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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
−17.36075450000251963255343060863, −16.56884225489264724970550555241, −14.46939174443194527204088265586, −13.25027207263843350395263492753, −11.40781153803159857100215676377, −9.936270644191942781225122439573, −9.144255972555358989072155479188, −7.09693738861898260641749473988, −5.22421898438878262057715007347, −1.37484034225738829292120101458,
1.37484034225738829292120101458, 5.22421898438878262057715007347, 7.09693738861898260641749473988, 9.144255972555358989072155479188, 9.936270644191942781225122439573, 11.40781153803159857100215676377, 13.25027207263843350395263492753, 14.46939174443194527204088265586, 16.56884225489264724970550555241, 17.36075450000251963255343060863
