| L(s) = 1 | − 0.765·2-s + 3-s − 1.41·4-s − 3.34·5-s − 0.765·6-s + 7-s + 2.61·8-s + 9-s + 2.56·10-s − 2.72·11-s − 1.41·12-s + 2.91·13-s − 0.765·14-s − 3.34·15-s + 0.823·16-s − 1.11·17-s − 0.765·18-s − 1.90·19-s + 4.72·20-s + 21-s + 2.08·22-s − 3.99·23-s + 2.61·24-s + 6.18·25-s − 2.23·26-s + 27-s − 1.41·28-s + ⋯ |
| L(s) = 1 | − 0.541·2-s + 0.577·3-s − 0.706·4-s − 1.49·5-s − 0.312·6-s + 0.377·7-s + 0.924·8-s + 0.333·9-s + 0.810·10-s − 0.820·11-s − 0.407·12-s + 0.808·13-s − 0.204·14-s − 0.863·15-s + 0.205·16-s − 0.270·17-s − 0.180·18-s − 0.436·19-s + 1.05·20-s + 0.218·21-s + 0.444·22-s − 0.832·23-s + 0.533·24-s + 1.23·25-s − 0.437·26-s + 0.192·27-s − 0.267·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 + 0.765T + 2T^{2} \) |
| 5 | \( 1 + 3.34T + 5T^{2} \) |
| 11 | \( 1 + 2.72T + 11T^{2} \) |
| 13 | \( 1 - 2.91T + 13T^{2} \) |
| 17 | \( 1 + 1.11T + 17T^{2} \) |
| 19 | \( 1 + 1.90T + 19T^{2} \) |
| 23 | \( 1 + 3.99T + 23T^{2} \) |
| 29 | \( 1 + 5.21T + 29T^{2} \) |
| 31 | \( 1 - 4.59T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 + 4.17T + 41T^{2} \) |
| 43 | \( 1 + 2.49T + 43T^{2} \) |
| 47 | \( 1 - 7.33T + 47T^{2} \) |
| 53 | \( 1 - 9.56T + 53T^{2} \) |
| 59 | \( 1 + 1.70T + 59T^{2} \) |
| 61 | \( 1 - 6.64T + 61T^{2} \) |
| 67 | \( 1 + 5.44T + 67T^{2} \) |
| 71 | \( 1 + 7.55T + 71T^{2} \) |
| 73 | \( 1 + 9.55T + 73T^{2} \) |
| 79 | \( 1 - 6.37T + 79T^{2} \) |
| 83 | \( 1 + 2.15T + 83T^{2} \) |
| 89 | \( 1 - 8.52T + 89T^{2} \) |
| 97 | \( 1 + 3.69T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.65705135495204555678345886411, −7.30472475450782724055624887734, −6.15556734487625277811907754439, −5.23052119376531147998605357015, −4.27710623374551790402148582272, −4.11739352887950225906464572615, −3.23364548314721950618253989663, −2.18242600103023065304414815879, −0.997360018639259512475813755335, 0,
0.997360018639259512475813755335, 2.18242600103023065304414815879, 3.23364548314721950618253989663, 4.11739352887950225906464572615, 4.27710623374551790402148582272, 5.23052119376531147998605357015, 6.15556734487625277811907754439, 7.30472475450782724055624887734, 7.65705135495204555678345886411
