L(s) = 1 | − 4.46·2-s + 11.9·4-s − 5·5-s − 17.6·8-s + 22.3·10-s + 56.5·11-s − 40.9·13-s − 16.6·16-s + 2.18·17-s − 16.4·19-s − 59.7·20-s − 252.·22-s + 155.·23-s + 25·25-s + 183.·26-s + 6.26·29-s − 168.·31-s + 215.·32-s − 9.77·34-s − 37.1·37-s + 73.5·38-s + 88.4·40-s − 266.·41-s − 14.6·43-s + 676.·44-s − 693.·46-s − 169.·47-s + ⋯ |
L(s) = 1 | − 1.57·2-s + 1.49·4-s − 0.447·5-s − 0.781·8-s + 0.706·10-s + 1.54·11-s − 0.873·13-s − 0.260·16-s + 0.0312·17-s − 0.198·19-s − 0.668·20-s − 2.44·22-s + 1.40·23-s + 0.200·25-s + 1.38·26-s + 0.0400·29-s − 0.977·31-s + 1.19·32-s − 0.0493·34-s − 0.165·37-s + 0.314·38-s + 0.349·40-s − 1.01·41-s − 0.0519·43-s + 2.31·44-s − 2.22·46-s − 0.527·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 4.46T + 8T^{2} \) |
| 11 | \( 1 - 56.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 40.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 2.18T + 4.91e3T^{2} \) |
| 19 | \( 1 + 16.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 155.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 6.26T + 2.43e4T^{2} \) |
| 31 | \( 1 + 168.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 37.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 266.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 14.6T + 7.95e4T^{2} \) |
| 47 | \( 1 + 169.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 151.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 234.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 242.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 820.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 961.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 934.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 300.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.08e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.12e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 752.T + 9.12e5T^{2} \) |
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
−8.577648669701098265484375790233, −7.58409654765888915337872268977, −7.03094791753694980624143862332, −6.45472420143965209062030297420, −5.13839343490042737674841580871, −4.17027690025449907385227268342, −3.10807392973969823565264596720, −1.89957176120048948966404910956, −1.03782097414346143926121900093, 0,
1.03782097414346143926121900093, 1.89957176120048948966404910956, 3.10807392973969823565264596720, 4.17027690025449907385227268342, 5.13839343490042737674841580871, 6.45472420143965209062030297420, 7.03094791753694980624143862332, 7.58409654765888915337872268977, 8.577648669701098265484375790233