L(s) = 1 | + 2-s + 0.731·3-s + 4-s + 3.24·5-s + 0.731·6-s + 4.02·7-s + 8-s − 2.46·9-s + 3.24·10-s + 1.64·11-s + 0.731·12-s + 2.18·13-s + 4.02·14-s + 2.37·15-s + 16-s − 2.11·17-s − 2.46·18-s − 19-s + 3.24·20-s + 2.94·21-s + 1.64·22-s + 7.93·23-s + 0.731·24-s + 5.50·25-s + 2.18·26-s − 3.99·27-s + 4.02·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.422·3-s + 0.5·4-s + 1.44·5-s + 0.298·6-s + 1.52·7-s + 0.353·8-s − 0.821·9-s + 1.02·10-s + 0.495·11-s + 0.211·12-s + 0.605·13-s + 1.07·14-s + 0.612·15-s + 0.250·16-s − 0.513·17-s − 0.581·18-s − 0.229·19-s + 0.724·20-s + 0.642·21-s + 0.350·22-s + 1.65·23-s + 0.149·24-s + 1.10·25-s + 0.428·26-s − 0.769·27-s + 0.760·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.463126862\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.463126862\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
good | 3 | \( 1 - 0.731T + 3T^{2} \) |
| 5 | \( 1 - 3.24T + 5T^{2} \) |
| 7 | \( 1 - 4.02T + 7T^{2} \) |
| 11 | \( 1 - 1.64T + 11T^{2} \) |
| 13 | \( 1 - 2.18T + 13T^{2} \) |
| 17 | \( 1 + 2.11T + 17T^{2} \) |
| 23 | \( 1 - 7.93T + 23T^{2} \) |
| 29 | \( 1 + 7.04T + 29T^{2} \) |
| 31 | \( 1 - 2.96T + 31T^{2} \) |
| 37 | \( 1 + 1.09T + 37T^{2} \) |
| 41 | \( 1 - 5.86T + 41T^{2} \) |
| 43 | \( 1 + 10.0T + 43T^{2} \) |
| 47 | \( 1 - 7.44T + 47T^{2} \) |
| 53 | \( 1 + 1.80T + 53T^{2} \) |
| 59 | \( 1 + 8.17T + 59T^{2} \) |
| 61 | \( 1 + 9.55T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 - 2.43T + 71T^{2} \) |
| 73 | \( 1 - 0.129T + 73T^{2} \) |
| 79 | \( 1 - 2.09T + 79T^{2} \) |
| 83 | \( 1 + 1.41T + 83T^{2} \) |
| 89 | \( 1 + 7.58T + 89T^{2} \) |
| 97 | \( 1 - 1.83T + 97T^{2} \) |
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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
−7.84660463721182582191571588155, −7.01563963535246591940072472735, −6.25544118944319891227482188269, −5.66058417693321389614771362873, −5.10578480488433577472634518266, −4.44035295189510568599647423975, −3.45611560333857422259846778930, −2.58508655660716549296240168194, −1.90679946457608528662640672253, −1.25413804884885295908263116030,
1.25413804884885295908263116030, 1.90679946457608528662640672253, 2.58508655660716549296240168194, 3.45611560333857422259846778930, 4.44035295189510568599647423975, 5.10578480488433577472634518266, 5.66058417693321389614771362873, 6.25544118944319891227482188269, 7.01563963535246591940072472735, 7.84660463721182582191571588155