| L(s) = 1 | + 1.56·2-s − 3-s + 0.438·4-s + 5-s − 1.56·6-s − 7-s − 2.43·8-s + 9-s + 1.56·10-s + 11-s − 0.438·12-s − 7.12·13-s − 1.56·14-s − 15-s − 4.68·16-s + 0.561·17-s + 1.56·18-s − 2.56·19-s + 0.438·20-s + 21-s + 1.56·22-s − 1.43·23-s + 2.43·24-s + 25-s − 11.1·26-s − 27-s − 0.438·28-s + ⋯ |
| L(s) = 1 | + 1.10·2-s − 0.577·3-s + 0.219·4-s + 0.447·5-s − 0.637·6-s − 0.377·7-s − 0.862·8-s + 0.333·9-s + 0.493·10-s + 0.301·11-s − 0.126·12-s − 1.97·13-s − 0.417·14-s − 0.258·15-s − 1.17·16-s + 0.136·17-s + 0.368·18-s − 0.587·19-s + 0.0980·20-s + 0.218·21-s + 0.332·22-s − 0.299·23-s + 0.497·24-s + 0.200·25-s − 2.18·26-s − 0.192·27-s − 0.0828·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 1.56T + 2T^{2} \) |
| 13 | \( 1 + 7.12T + 13T^{2} \) |
| 17 | \( 1 - 0.561T + 17T^{2} \) |
| 19 | \( 1 + 2.56T + 19T^{2} \) |
| 23 | \( 1 + 1.43T + 23T^{2} \) |
| 29 | \( 1 - 1.68T + 29T^{2} \) |
| 31 | \( 1 + 5.12T + 31T^{2} \) |
| 37 | \( 1 + 7.12T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 2.56T + 43T^{2} \) |
| 47 | \( 1 + 5.12T + 47T^{2} \) |
| 53 | \( 1 + 13.6T + 53T^{2} \) |
| 59 | \( 1 - 10.5T + 59T^{2} \) |
| 61 | \( 1 + 3.43T + 61T^{2} \) |
| 67 | \( 1 - 6.87T + 67T^{2} \) |
| 71 | \( 1 - 5.12T + 71T^{2} \) |
| 73 | \( 1 + 11.1T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 + 5.43T + 83T^{2} \) |
| 89 | \( 1 + 2.31T + 89T^{2} \) |
| 97 | \( 1 - 2.80T + 97T^{2} \) |
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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
−9.619147841225628573576640357938, −8.693753344415080044023494634834, −7.36721932825129260434292650659, −6.62439192388111697068340986083, −5.79612347605071384668189312692, −5.05129384442759653922738385098, −4.37273487966080637528632387399, −3.23581817406617446204227935853, −2.12850837763730799097357776973, 0,
2.12850837763730799097357776973, 3.23581817406617446204227935853, 4.37273487966080637528632387399, 5.05129384442759653922738385098, 5.79612347605071384668189312692, 6.62439192388111697068340986083, 7.36721932825129260434292650659, 8.693753344415080044023494634834, 9.619147841225628573576640357938
