| L(s) = 1 | + 2-s − 3-s + 4-s + 4.15·5-s − 6-s + 0.461·7-s + 8-s + 9-s + 4.15·10-s + 2.29·11-s − 12-s − 2.16·13-s + 0.461·14-s − 4.15·15-s + 16-s + 3.68·17-s + 18-s + 0.461·19-s + 4.15·20-s − 0.461·21-s + 2.29·22-s + 23-s − 24-s + 12.2·25-s − 2.16·26-s − 27-s + 0.461·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.85·5-s − 0.408·6-s + 0.174·7-s + 0.353·8-s + 0.333·9-s + 1.31·10-s + 0.691·11-s − 0.288·12-s − 0.600·13-s + 0.123·14-s − 1.07·15-s + 0.250·16-s + 0.894·17-s + 0.235·18-s + 0.105·19-s + 0.929·20-s − 0.100·21-s + 0.488·22-s + 0.208·23-s − 0.204·24-s + 2.45·25-s − 0.424·26-s − 0.192·27-s + 0.0872·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.037134488\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.037134488\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 - 4.15T + 5T^{2} \) |
| 7 | \( 1 - 0.461T + 7T^{2} \) |
| 11 | \( 1 - 2.29T + 11T^{2} \) |
| 13 | \( 1 + 2.16T + 13T^{2} \) |
| 17 | \( 1 - 3.68T + 17T^{2} \) |
| 19 | \( 1 - 0.461T + 19T^{2} \) |
| 31 | \( 1 + 2.61T + 31T^{2} \) |
| 37 | \( 1 + 7.35T + 37T^{2} \) |
| 41 | \( 1 - 1.98T + 41T^{2} \) |
| 43 | \( 1 + 4.73T + 43T^{2} \) |
| 47 | \( 1 - 12.4T + 47T^{2} \) |
| 53 | \( 1 + 7.91T + 53T^{2} \) |
| 59 | \( 1 - 13.9T + 59T^{2} \) |
| 61 | \( 1 - 7.56T + 61T^{2} \) |
| 67 | \( 1 + 1.07T + 67T^{2} \) |
| 71 | \( 1 + 7.96T + 71T^{2} \) |
| 73 | \( 1 - 14.6T + 73T^{2} \) |
| 79 | \( 1 + 16.1T + 79T^{2} \) |
| 83 | \( 1 - 1.22T + 83T^{2} \) |
| 89 | \( 1 - 1.14T + 89T^{2} \) |
| 97 | \( 1 + 13.6T + 97T^{2} \) |
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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
−8.550472489428161534067156541147, −7.35747533725142878019463852775, −6.72816406049138503336846695626, −6.10730896872295301801781009240, −5.35152557775888084287255889485, −5.09107484140501474947857933616, −3.95391306704125882280871903823, −2.88268149853655992603567265294, −1.96223556634589965983488624934, −1.18551957697084490724709988211,
1.18551957697084490724709988211, 1.96223556634589965983488624934, 2.88268149853655992603567265294, 3.95391306704125882280871903823, 5.09107484140501474947857933616, 5.35152557775888084287255889485, 6.10730896872295301801781009240, 6.72816406049138503336846695626, 7.35747533725142878019463852775, 8.550472489428161534067156541147
