| L(s) = 1 | − 0.634·2-s − 1.50·3-s − 1.59·4-s − 1.64·5-s + 0.955·6-s − 3.41·7-s + 2.28·8-s − 0.727·9-s + 1.04·10-s − 11-s + 2.40·12-s − 1.04·13-s + 2.16·14-s + 2.48·15-s + 1.74·16-s − 7.31·17-s + 0.461·18-s − 7.44·19-s + 2.62·20-s + 5.14·21-s + 0.634·22-s + 5.97·23-s − 3.43·24-s − 2.29·25-s + 0.659·26-s + 5.61·27-s + 5.45·28-s + ⋯ |
| L(s) = 1 | − 0.448·2-s − 0.870·3-s − 0.798·4-s − 0.735·5-s + 0.390·6-s − 1.28·7-s + 0.806·8-s − 0.242·9-s + 0.329·10-s − 0.301·11-s + 0.695·12-s − 0.288·13-s + 0.578·14-s + 0.640·15-s + 0.437·16-s − 1.77·17-s + 0.108·18-s − 1.70·19-s + 0.587·20-s + 1.12·21-s + 0.135·22-s + 1.24·23-s − 0.702·24-s − 0.458·25-s + 0.129·26-s + 1.08·27-s + 1.03·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1856500447\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1856500447\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 + 0.634T + 2T^{2} \) |
| 3 | \( 1 + 1.50T + 3T^{2} \) |
| 5 | \( 1 + 1.64T + 5T^{2} \) |
| 7 | \( 1 + 3.41T + 7T^{2} \) |
| 13 | \( 1 + 1.04T + 13T^{2} \) |
| 17 | \( 1 + 7.31T + 17T^{2} \) |
| 19 | \( 1 + 7.44T + 19T^{2} \) |
| 23 | \( 1 - 5.97T + 23T^{2} \) |
| 29 | \( 1 - 9.96T + 29T^{2} \) |
| 31 | \( 1 - 1.32T + 31T^{2} \) |
| 37 | \( 1 - 7.10T + 37T^{2} \) |
| 41 | \( 1 + 0.452T + 41T^{2} \) |
| 43 | \( 1 - 6.46T + 43T^{2} \) |
| 47 | \( 1 + 12.9T + 47T^{2} \) |
| 53 | \( 1 + 0.807T + 53T^{2} \) |
| 59 | \( 1 + 4.34T + 59T^{2} \) |
| 67 | \( 1 + 6.57T + 67T^{2} \) |
| 71 | \( 1 - 6.78T + 71T^{2} \) |
| 73 | \( 1 - 3.97T + 73T^{2} \) |
| 79 | \( 1 - 1.36T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 - 15.3T + 89T^{2} \) |
| 97 | \( 1 - 0.131T + 97T^{2} \) |
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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
−10.60545532490035497118656064871, −9.626398533405362576429605609060, −8.791746303373136901022388614319, −8.111989075519809906317949003503, −6.77820424730230162742733745914, −6.24864997288576982069967666444, −4.86100019714922748526519458562, −4.21523213032346807070285691410, −2.79498680456650776647268131644, −0.38709172973278171767988495491,
0.38709172973278171767988495491, 2.79498680456650776647268131644, 4.21523213032346807070285691410, 4.86100019714922748526519458562, 6.24864997288576982069967666444, 6.77820424730230162742733745914, 8.111989075519809906317949003503, 8.791746303373136901022388614319, 9.626398533405362576429605609060, 10.60545532490035497118656064871
