| L(s) = 1 | + 0.401·2-s + 3-s − 1.83·4-s + 5-s + 0.401·6-s − 7-s − 1.54·8-s + 9-s + 0.401·10-s − 1.58·11-s − 1.83·12-s + 0.690·13-s − 0.401·14-s + 15-s + 3.05·16-s + 17-s + 0.401·18-s + 7.52·19-s − 1.83·20-s − 21-s − 0.637·22-s − 5.73·23-s − 1.54·24-s + 25-s + 0.277·26-s + 27-s + 1.83·28-s + ⋯ |
| L(s) = 1 | + 0.283·2-s + 0.577·3-s − 0.919·4-s + 0.447·5-s + 0.163·6-s − 0.377·7-s − 0.544·8-s + 0.333·9-s + 0.126·10-s − 0.478·11-s − 0.530·12-s + 0.191·13-s − 0.107·14-s + 0.258·15-s + 0.764·16-s + 0.242·17-s + 0.0945·18-s + 1.72·19-s − 0.411·20-s − 0.218·21-s − 0.135·22-s − 1.19·23-s − 0.314·24-s + 0.200·25-s + 0.0543·26-s + 0.192·27-s + 0.347·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1785 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1785 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.033877141\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.033877141\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 - 0.401T + 2T^{2} \) |
| 11 | \( 1 + 1.58T + 11T^{2} \) |
| 13 | \( 1 - 0.690T + 13T^{2} \) |
| 19 | \( 1 - 7.52T + 19T^{2} \) |
| 23 | \( 1 + 5.73T + 23T^{2} \) |
| 29 | \( 1 - 4.89T + 29T^{2} \) |
| 31 | \( 1 - 6.03T + 31T^{2} \) |
| 37 | \( 1 + 7.73T + 37T^{2} \) |
| 41 | \( 1 - 2.44T + 41T^{2} \) |
| 43 | \( 1 - 0.843T + 43T^{2} \) |
| 47 | \( 1 - 8.89T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 - 4.84T + 61T^{2} \) |
| 67 | \( 1 - 5.50T + 67T^{2} \) |
| 71 | \( 1 + 12.6T + 71T^{2} \) |
| 73 | \( 1 + 7.13T + 73T^{2} \) |
| 79 | \( 1 - 3.34T + 79T^{2} \) |
| 83 | \( 1 - 1.01T + 83T^{2} \) |
| 89 | \( 1 - 10.1T + 89T^{2} \) |
| 97 | \( 1 + 3.02T + 97T^{2} \) |
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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
−9.260214263500005760493817364401, −8.576103790797159591998871800223, −7.86379338093784380705684842964, −6.94722511139056692699607777694, −5.82688000098945227118790394662, −5.26481081093917156477312744136, −4.22507009248878371991583270198, −3.39154210382621853266414281170, −2.51320270100991822290895152880, −0.945408199273842509748456111625,
0.945408199273842509748456111625, 2.51320270100991822290895152880, 3.39154210382621853266414281170, 4.22507009248878371991583270198, 5.26481081093917156477312744136, 5.82688000098945227118790394662, 6.94722511139056692699607777694, 7.86379338093784380705684842964, 8.576103790797159591998871800223, 9.260214263500005760493817364401
