| L(s) = 1 | − 2.66·2-s − 1.09·3-s + 5.08·4-s + 3.59·5-s + 2.92·6-s + 1.69·7-s − 8.21·8-s − 1.79·9-s − 9.56·10-s − 5.58·12-s − 0.854·13-s − 4.50·14-s − 3.94·15-s + 11.6·16-s − 7.31·17-s + 4.78·18-s + 5.68·19-s + 18.2·20-s − 1.85·21-s + 0.333·23-s + 9.01·24-s + 7.90·25-s + 2.27·26-s + 5.26·27-s + 8.61·28-s + 3.09·29-s + 10.4·30-s + ⋯ |
| L(s) = 1 | − 1.88·2-s − 0.633·3-s + 2.54·4-s + 1.60·5-s + 1.19·6-s + 0.639·7-s − 2.90·8-s − 0.598·9-s − 3.02·10-s − 1.61·12-s − 0.237·13-s − 1.20·14-s − 1.01·15-s + 2.92·16-s − 1.77·17-s + 1.12·18-s + 1.30·19-s + 4.08·20-s − 0.405·21-s + 0.0695·23-s + 1.84·24-s + 1.58·25-s + 0.446·26-s + 1.01·27-s + 1.62·28-s + 0.574·29-s + 1.91·30-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
| \(L(1)\) |
\(\approx\) |
\(0.7440215086\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7440215086\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.66T + 2T^{2} \) |
| 3 | \( 1 + 1.09T + 3T^{2} \) |
| 5 | \( 1 - 3.59T + 5T^{2} \) |
| 7 | \( 1 - 1.69T + 7T^{2} \) |
| 13 | \( 1 + 0.854T + 13T^{2} \) |
| 17 | \( 1 + 7.31T + 17T^{2} \) |
| 19 | \( 1 - 5.68T + 19T^{2} \) |
| 23 | \( 1 - 0.333T + 23T^{2} \) |
| 29 | \( 1 - 3.09T + 29T^{2} \) |
| 31 | \( 1 + 2.66T + 31T^{2} \) |
| 37 | \( 1 - 6.26T + 37T^{2} \) |
| 41 | \( 1 - 9.38T + 41T^{2} \) |
| 43 | \( 1 - 0.315T + 43T^{2} \) |
| 47 | \( 1 - 0.371T + 47T^{2} \) |
| 53 | \( 1 - 3.46T + 53T^{2} \) |
| 59 | \( 1 + 3.29T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 + 9.33T + 67T^{2} \) |
| 71 | \( 1 - 4.51T + 71T^{2} \) |
| 73 | \( 1 - 6.74T + 73T^{2} \) |
| 79 | \( 1 - 13.0T + 79T^{2} \) |
| 83 | \( 1 + 9.65T + 83T^{2} \) |
| 89 | \( 1 - 5.82T + 89T^{2} \) |
| 97 | \( 1 - 17.8T + 97T^{2} \) |
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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
−9.485516796884708035503033556073, −9.053350963490474783552260394161, −8.247652696380771704712495005579, −7.26462088978125206000606583358, −6.42837293576010156419251781181, −5.85830538445827258802068688010, −4.93124837120190142560907796333, −2.71325823345896117528890745833, −2.01796650566119242405380983345, −0.865370235832720628333107786519,
0.865370235832720628333107786519, 2.01796650566119242405380983345, 2.71325823345896117528890745833, 4.93124837120190142560907796333, 5.85830538445827258802068688010, 6.42837293576010156419251781181, 7.26462088978125206000606583358, 8.247652696380771704712495005579, 9.053350963490474783552260394161, 9.485516796884708035503033556073
