| L(s) = 1 | + 2-s + 3-s + 4-s + 2.75·5-s + 6-s + 1.74·7-s + 8-s + 9-s + 2.75·10-s + 6.11·11-s + 12-s − 5.30·13-s + 1.74·14-s + 2.75·15-s + 16-s + 17-s + 18-s + 4.22·19-s + 2.75·20-s + 1.74·21-s + 6.11·22-s − 7.05·23-s + 24-s + 2.59·25-s − 5.30·26-s + 27-s + 1.74·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.23·5-s + 0.408·6-s + 0.660·7-s + 0.353·8-s + 0.333·9-s + 0.871·10-s + 1.84·11-s + 0.288·12-s − 1.47·13-s + 0.467·14-s + 0.711·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s + 0.969·19-s + 0.616·20-s + 0.381·21-s + 1.30·22-s − 1.47·23-s + 0.204·24-s + 0.518·25-s − 1.04·26-s + 0.192·27-s + 0.330·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.068629219\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.068629219\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| good | 5 | \( 1 - 2.75T + 5T^{2} \) |
| 7 | \( 1 - 1.74T + 7T^{2} \) |
| 11 | \( 1 - 6.11T + 11T^{2} \) |
| 13 | \( 1 + 5.30T + 13T^{2} \) |
| 19 | \( 1 - 4.22T + 19T^{2} \) |
| 23 | \( 1 + 7.05T + 23T^{2} \) |
| 29 | \( 1 + 2.85T + 29T^{2} \) |
| 31 | \( 1 + 0.451T + 31T^{2} \) |
| 37 | \( 1 + 2.31T + 37T^{2} \) |
| 41 | \( 1 + 4.96T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 - 8.55T + 53T^{2} \) |
| 61 | \( 1 + 7.55T + 61T^{2} \) |
| 67 | \( 1 - 5.93T + 67T^{2} \) |
| 71 | \( 1 - 7.26T + 71T^{2} \) |
| 73 | \( 1 + 4.37T + 73T^{2} \) |
| 79 | \( 1 - 7.90T + 79T^{2} \) |
| 83 | \( 1 + 4.90T + 83T^{2} \) |
| 89 | \( 1 + 14.0T + 89T^{2} \) |
| 97 | \( 1 + 3.80T + 97T^{2} \) |
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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
−7.929085022325845748872614768231, −7.26948913101139846608159382484, −6.64089483396118544959147598059, −5.77602688259506485827264064875, −5.31327485567532015142806365806, −4.32959680200368051954936062012, −3.79249690440462629593796573015, −2.66572777947043540197231051866, −1.98321850532836136921590415939, −1.29545166816114143478957564289,
1.29545166816114143478957564289, 1.98321850532836136921590415939, 2.66572777947043540197231051866, 3.79249690440462629593796573015, 4.32959680200368051954936062012, 5.31327485567532015142806365806, 5.77602688259506485827264064875, 6.64089483396118544959147598059, 7.26948913101139846608159382484, 7.929085022325845748872614768231
