| L(s) = 1 | + 0.648·2-s − 3-s − 1.57·4-s − 0.569·5-s − 0.648·6-s − 7-s − 2.32·8-s + 9-s − 0.369·10-s + 2.97·11-s + 1.57·12-s − 4.47·13-s − 0.648·14-s + 0.569·15-s + 1.65·16-s + 0.912·17-s + 0.648·18-s + 7.98·19-s + 0.900·20-s + 21-s + 1.92·22-s − 3.22·23-s + 2.32·24-s − 4.67·25-s − 2.89·26-s − 27-s + 1.57·28-s + ⋯ |
| L(s) = 1 | + 0.458·2-s − 0.577·3-s − 0.789·4-s − 0.254·5-s − 0.264·6-s − 0.377·7-s − 0.820·8-s + 0.333·9-s − 0.116·10-s + 0.896·11-s + 0.456·12-s − 1.24·13-s − 0.173·14-s + 0.147·15-s + 0.414·16-s + 0.221·17-s + 0.152·18-s + 1.83·19-s + 0.201·20-s + 0.218·21-s + 0.411·22-s − 0.673·23-s + 0.473·24-s − 0.935·25-s − 0.568·26-s − 0.192·27-s + 0.298·28-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
| \(L(1)\) |
\(\approx\) |
\(0.9153605786\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9153605786\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.648T + 2T^{2} \) |
| 5 | \( 1 + 0.569T + 5T^{2} \) |
| 11 | \( 1 - 2.97T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 - 0.912T + 17T^{2} \) |
| 19 | \( 1 - 7.98T + 19T^{2} \) |
| 23 | \( 1 + 3.22T + 23T^{2} \) |
| 29 | \( 1 + 7.97T + 29T^{2} \) |
| 31 | \( 1 - 3.42T + 31T^{2} \) |
| 37 | \( 1 + 0.0173T + 37T^{2} \) |
| 41 | \( 1 + 9.33T + 41T^{2} \) |
| 43 | \( 1 + 2.94T + 43T^{2} \) |
| 47 | \( 1 - 2.15T + 47T^{2} \) |
| 53 | \( 1 - 4.27T + 53T^{2} \) |
| 59 | \( 1 + 5.28T + 59T^{2} \) |
| 61 | \( 1 + 4.90T + 61T^{2} \) |
| 67 | \( 1 - 11.9T + 67T^{2} \) |
| 71 | \( 1 - 2.36T + 71T^{2} \) |
| 73 | \( 1 - 4.45T + 73T^{2} \) |
| 79 | \( 1 + 4.96T + 79T^{2} \) |
| 83 | \( 1 + 5.20T + 83T^{2} \) |
| 89 | \( 1 + 15.1T + 89T^{2} \) |
| 97 | \( 1 + 17.6T + 97T^{2} \) |
| 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
−7.68482487686500908058059868318, −7.12968373377343305530969794795, −6.31687036858269008554283913618, −5.50909504994592592052355065720, −5.18165513124682018384173276186, −4.25071045482821827812935408377, −3.71011096272745764198840952262, −2.95905397973814046129397285277, −1.65033287815609233708107623710, −0.46185630450591570494890699881,
0.46185630450591570494890699881, 1.65033287815609233708107623710, 2.95905397973814046129397285277, 3.71011096272745764198840952262, 4.25071045482821827812935408377, 5.18165513124682018384173276186, 5.50909504994592592052355065720, 6.31687036858269008554283913618, 7.12968373377343305530969794795, 7.68482487686500908058059868318
