| L(s) = 1 | + 2.13·2-s + 2.50·3-s + 2.56·4-s − 0.997·5-s + 5.35·6-s − 0.526·7-s + 1.20·8-s + 3.29·9-s − 2.13·10-s + 11-s + 6.43·12-s + 5.55·13-s − 1.12·14-s − 2.50·15-s − 2.55·16-s − 5.19·17-s + 7.03·18-s − 4.57·19-s − 2.55·20-s − 1.32·21-s + 2.13·22-s + 7.05·23-s + 3.02·24-s − 4.00·25-s + 11.8·26-s + 0.735·27-s − 1.35·28-s + ⋯ |
| L(s) = 1 | + 1.51·2-s + 1.44·3-s + 1.28·4-s − 0.446·5-s + 2.18·6-s − 0.199·7-s + 0.425·8-s + 1.09·9-s − 0.674·10-s + 0.301·11-s + 1.85·12-s + 1.54·13-s − 0.300·14-s − 0.646·15-s − 0.638·16-s − 1.26·17-s + 1.65·18-s − 1.05·19-s − 0.572·20-s − 0.288·21-s + 0.455·22-s + 1.47·23-s + 0.616·24-s − 0.800·25-s + 2.32·26-s + 0.141·27-s − 0.255·28-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
| \(L(1)\) |
\(\approx\) |
\(4.620151284\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.620151284\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.13T + 2T^{2} \) |
| 3 | \( 1 - 2.50T + 3T^{2} \) |
| 5 | \( 1 + 0.997T + 5T^{2} \) |
| 7 | \( 1 + 0.526T + 7T^{2} \) |
| 13 | \( 1 - 5.55T + 13T^{2} \) |
| 17 | \( 1 + 5.19T + 17T^{2} \) |
| 19 | \( 1 + 4.57T + 19T^{2} \) |
| 23 | \( 1 - 7.05T + 23T^{2} \) |
| 29 | \( 1 - 1.54T + 29T^{2} \) |
| 31 | \( 1 + 7.26T + 31T^{2} \) |
| 37 | \( 1 - 5.01T + 37T^{2} \) |
| 41 | \( 1 - 3.84T + 41T^{2} \) |
| 43 | \( 1 - 9.71T + 43T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 + 0.817T + 53T^{2} \) |
| 59 | \( 1 + 9.14T + 59T^{2} \) |
| 67 | \( 1 - 12.5T + 67T^{2} \) |
| 71 | \( 1 + 6.89T + 71T^{2} \) |
| 73 | \( 1 - 9.04T + 73T^{2} \) |
| 79 | \( 1 + 16.0T + 79T^{2} \) |
| 83 | \( 1 - 16.4T + 83T^{2} \) |
| 89 | \( 1 - 12.0T + 89T^{2} \) |
| 97 | \( 1 - 11.2T + 97T^{2} \) |
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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
−10.96624958468882092618602917263, −9.272496149147951950301366190670, −8.836472766872080862039938352300, −7.88173761905525769018491557663, −6.76260367519403166778630149342, −6.02295925170488602990690346552, −4.56988359031613720919312153679, −3.85097294813388732796867050116, −3.17155576733440516643119994314, −2.05721171043202889489709778487,
2.05721171043202889489709778487, 3.17155576733440516643119994314, 3.85097294813388732796867050116, 4.56988359031613720919312153679, 6.02295925170488602990690346552, 6.76260367519403166778630149342, 7.88173761905525769018491557663, 8.836472766872080862039938352300, 9.272496149147951950301366190670, 10.96624958468882092618602917263
