| L(s) = 1 | + 2-s + 1.37·3-s + 4-s + 3.53·5-s + 1.37·6-s − 2.08·7-s + 8-s − 1.10·9-s + 3.53·10-s − 3.14·11-s + 1.37·12-s − 3.99·13-s − 2.08·14-s + 4.87·15-s + 16-s − 6.01·17-s − 1.10·18-s + 19-s + 3.53·20-s − 2.87·21-s − 3.14·22-s − 6.22·23-s + 1.37·24-s + 7.51·25-s − 3.99·26-s − 5.65·27-s − 2.08·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.795·3-s + 0.5·4-s + 1.58·5-s + 0.562·6-s − 0.788·7-s + 0.353·8-s − 0.367·9-s + 1.11·10-s − 0.947·11-s + 0.397·12-s − 1.10·13-s − 0.557·14-s + 1.25·15-s + 0.250·16-s − 1.45·17-s − 0.259·18-s + 0.229·19-s + 0.790·20-s − 0.626·21-s − 0.670·22-s − 1.29·23-s + 0.281·24-s + 1.50·25-s − 0.783·26-s − 1.08·27-s − 0.394·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 - 1.37T + 3T^{2} \) |
| 5 | \( 1 - 3.53T + 5T^{2} \) |
| 7 | \( 1 + 2.08T + 7T^{2} \) |
| 11 | \( 1 + 3.14T + 11T^{2} \) |
| 13 | \( 1 + 3.99T + 13T^{2} \) |
| 17 | \( 1 + 6.01T + 17T^{2} \) |
| 23 | \( 1 + 6.22T + 23T^{2} \) |
| 29 | \( 1 + 6.86T + 29T^{2} \) |
| 31 | \( 1 - 6.59T + 31T^{2} \) |
| 37 | \( 1 + 3.67T + 37T^{2} \) |
| 41 | \( 1 - 8.95T + 41T^{2} \) |
| 43 | \( 1 - 0.304T + 43T^{2} \) |
| 47 | \( 1 - 0.829T + 47T^{2} \) |
| 53 | \( 1 - 3.90T + 53T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 - 2.06T + 61T^{2} \) |
| 67 | \( 1 - 4.91T + 67T^{2} \) |
| 71 | \( 1 + 1.61T + 71T^{2} \) |
| 73 | \( 1 + 2.42T + 73T^{2} \) |
| 79 | \( 1 + 2.78T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 + 3.09T + 89T^{2} \) |
| 97 | \( 1 - 2.90T + 97T^{2} \) |
| 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
−7.44641789843387423648999679136, −6.58618155687846105968623307483, −6.03739356320367427493141288541, −5.46819933396954558633108704246, −4.75197346371912589098075668575, −3.82356804009347441759591375191, −2.77853141659500952639448813204, −2.47173287111427554462641741424, −1.88034744274486959387721623832, 0,
1.88034744274486959387721623832, 2.47173287111427554462641741424, 2.77853141659500952639448813204, 3.82356804009347441759591375191, 4.75197346371912589098075668575, 5.46819933396954558633108704246, 6.03739356320367427493141288541, 6.58618155687846105968623307483, 7.44641789843387423648999679136
