| L(s) = 1 | − 2-s − 3-s − 4-s + 2·5-s + 6-s + 3·8-s + 9-s − 2·10-s − 11-s + 12-s − 13-s − 2·15-s − 16-s + 6·17-s − 18-s + 4·19-s − 2·20-s + 22-s − 8·23-s − 3·24-s − 25-s + 26-s − 27-s − 10·29-s + 2·30-s − 5·32-s + 33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.894·5-s + 0.408·6-s + 1.06·8-s + 1/3·9-s − 0.632·10-s − 0.301·11-s + 0.288·12-s − 0.277·13-s − 0.516·15-s − 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.917·19-s − 0.447·20-s + 0.213·22-s − 1.66·23-s − 0.612·24-s − 1/5·25-s + 0.196·26-s − 0.192·27-s − 1.85·29-s + 0.365·30-s − 0.883·32-s + 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21021 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21021 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7929866532\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7929866532\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−15.88422811465214, −14.98361546285573, −14.43225125325289, −13.91028905786734, −13.45111311850996, −12.92519940999038, −12.30182402001567, −11.70906166770175, −11.12589267211235, −10.32769722453520, −9.944792000171638, −9.647449352094174, −9.150570534164287, −8.202206491835556, −7.715065259952207, −7.374125699116262, −6.329256602376267, −5.731652375824684, −5.366773491042070, −4.683585130464562, −3.854180423306234, −3.149147727525220, −1.910045576714515, −1.519793025995635, −0.4367548651366402,
0.4367548651366402, 1.519793025995635, 1.910045576714515, 3.149147727525220, 3.854180423306234, 4.683585130464562, 5.366773491042070, 5.731652375824684, 6.329256602376267, 7.374125699116262, 7.715065259952207, 8.202206491835556, 9.150570534164287, 9.647449352094174, 9.944792000171638, 10.32769722453520, 11.12589267211235, 11.70906166770175, 12.30182402001567, 12.92519940999038, 13.45111311850996, 13.91028905786734, 14.43225125325289, 14.98361546285573, 15.88422811465214
