| L(s) = 1 | − 2.41·2-s − 2.77·3-s + 3.85·4-s + 3.03·5-s + 6.70·6-s − 4.34·7-s − 4.48·8-s + 4.67·9-s − 7.33·10-s − 10.6·12-s − 2.36·13-s + 10.5·14-s − 8.39·15-s + 3.13·16-s + 3.94·17-s − 11.3·18-s + 0.986·19-s + 11.6·20-s + 12.0·21-s − 4.72·23-s + 12.4·24-s + 4.18·25-s + 5.71·26-s − 4.65·27-s − 16.7·28-s + 4.78·29-s + 20.3·30-s + ⋯ |
| L(s) = 1 | − 1.71·2-s − 1.59·3-s + 1.92·4-s + 1.35·5-s + 2.73·6-s − 1.64·7-s − 1.58·8-s + 1.55·9-s − 2.31·10-s − 3.08·12-s − 0.655·13-s + 2.80·14-s − 2.16·15-s + 0.783·16-s + 0.957·17-s − 2.66·18-s + 0.226·19-s + 2.61·20-s + 2.62·21-s − 0.985·23-s + 2.53·24-s + 0.836·25-s + 1.12·26-s − 0.895·27-s − 3.16·28-s + 0.889·29-s + 3.70·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3112892440\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3112892440\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 3 | \( 1 + 2.77T + 3T^{2} \) |
| 5 | \( 1 - 3.03T + 5T^{2} \) |
| 7 | \( 1 + 4.34T + 7T^{2} \) |
| 13 | \( 1 + 2.36T + 13T^{2} \) |
| 17 | \( 1 - 3.94T + 17T^{2} \) |
| 19 | \( 1 - 0.986T + 19T^{2} \) |
| 23 | \( 1 + 4.72T + 23T^{2} \) |
| 29 | \( 1 - 4.78T + 29T^{2} \) |
| 31 | \( 1 - 3.54T + 31T^{2} \) |
| 37 | \( 1 + 0.438T + 37T^{2} \) |
| 41 | \( 1 + 6.66T + 41T^{2} \) |
| 43 | \( 1 + 6.82T + 43T^{2} \) |
| 47 | \( 1 + 7.81T + 47T^{2} \) |
| 53 | \( 1 - 3.89T + 53T^{2} \) |
| 59 | \( 1 + 8.77T + 59T^{2} \) |
| 61 | \( 1 + 8.79T + 61T^{2} \) |
| 67 | \( 1 - 1.17T + 67T^{2} \) |
| 71 | \( 1 + 1.33T + 71T^{2} \) |
| 73 | \( 1 - 4.89T + 73T^{2} \) |
| 79 | \( 1 - 7.91T + 79T^{2} \) |
| 83 | \( 1 - 15.3T + 83T^{2} \) |
| 89 | \( 1 - 6.33T + 89T^{2} \) |
| 97 | \( 1 - 2.49T + 97T^{2} \) |
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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
−9.819662155671942422567990389517, −9.293822755904519908851342549614, −8.048740993227211296382143534054, −6.94921475326937766384450387568, −6.40494880430573825592155614365, −5.95874438299620660120369611084, −4.96698192881245995877870683805, −3.07714335923274823277349208389, −1.79569717866202016673890579876, −0.56052171561766806713078276291,
0.56052171561766806713078276291, 1.79569717866202016673890579876, 3.07714335923274823277349208389, 4.96698192881245995877870683805, 5.95874438299620660120369611084, 6.40494880430573825592155614365, 6.94921475326937766384450387568, 8.048740993227211296382143534054, 9.293822755904519908851342549614, 9.819662155671942422567990389517
