| L(s) = 1 | + 1.90·2-s + 3-s + 1.62·4-s + 1.90·6-s + 7-s − 0.719·8-s + 9-s + 2·11-s + 1.62·12-s + 6.42·13-s + 1.90·14-s − 4.61·16-s − 4.42·17-s + 1.90·18-s − 2.42·19-s + 21-s + 3.80·22-s − 1.37·23-s − 0.719·24-s + 12.2·26-s + 27-s + 1.62·28-s + 0.755·29-s + 5.18·31-s − 7.34·32-s + 2·33-s − 8.42·34-s + ⋯ |
| L(s) = 1 | + 1.34·2-s + 0.577·3-s + 0.811·4-s + 0.776·6-s + 0.377·7-s − 0.254·8-s + 0.333·9-s + 0.603·11-s + 0.468·12-s + 1.78·13-s + 0.508·14-s − 1.15·16-s − 1.07·17-s + 0.448·18-s − 0.557·19-s + 0.218·21-s + 0.811·22-s − 0.287·23-s − 0.146·24-s + 2.39·26-s + 0.192·27-s + 0.306·28-s + 0.140·29-s + 0.931·31-s − 1.29·32-s + 0.348·33-s − 1.44·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.398278472\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.398278472\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 2 | \( 1 - 1.90T + 2T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 6.42T + 13T^{2} \) |
| 17 | \( 1 + 4.42T + 17T^{2} \) |
| 19 | \( 1 + 2.42T + 19T^{2} \) |
| 23 | \( 1 + 1.37T + 23T^{2} \) |
| 29 | \( 1 - 0.755T + 29T^{2} \) |
| 31 | \( 1 - 5.18T + 31T^{2} \) |
| 37 | \( 1 + 7.61T + 37T^{2} \) |
| 41 | \( 1 + 8.23T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 + 2.75T + 47T^{2} \) |
| 53 | \( 1 + 9.18T + 53T^{2} \) |
| 59 | \( 1 - 14.1T + 59T^{2} \) |
| 61 | \( 1 - 6.85T + 61T^{2} \) |
| 67 | \( 1 + 2.75T + 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 - 1.57T + 73T^{2} \) |
| 79 | \( 1 + 4.85T + 79T^{2} \) |
| 83 | \( 1 + 11.6T + 83T^{2} \) |
| 89 | \( 1 - 4.62T + 89T^{2} \) |
| 97 | \( 1 - 11.9T + 97T^{2} \) |
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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
−11.22513960995139188819181077531, −10.09304373141097990113001522762, −8.709674419243524329560312130844, −8.486807367448386961998377543363, −6.79374606396769027039006230987, −6.24284072893515137486339851277, −4.98967227666984831662205492010, −4.05716625139708045112469583870, −3.31967611255563260991031607198, −1.85573080646300267740297235574,
1.85573080646300267740297235574, 3.31967611255563260991031607198, 4.05716625139708045112469583870, 4.98967227666984831662205492010, 6.24284072893515137486339851277, 6.79374606396769027039006230987, 8.486807367448386961998377543363, 8.709674419243524329560312130844, 10.09304373141097990113001522762, 11.22513960995139188819181077531
