| L(s) = 1 | − 1.92·2-s − 5.16·3-s − 4.27·4-s + 3.37·5-s + 9.95·6-s + 15.9·7-s + 23.6·8-s − 0.353·9-s − 6.51·10-s − 69.8·11-s + 22.0·12-s + 8.18·13-s − 30.8·14-s − 17.4·15-s − 11.4·16-s + 67.4·17-s + 0.682·18-s + 11.3·19-s − 14.4·20-s − 82.4·21-s + 134.·22-s + 140.·23-s − 122.·24-s − 113.·25-s − 15.7·26-s + 141.·27-s − 68.3·28-s + ⋯ |
| L(s) = 1 | − 0.682·2-s − 0.993·3-s − 0.534·4-s + 0.301·5-s + 0.677·6-s + 0.862·7-s + 1.04·8-s − 0.0131·9-s − 0.205·10-s − 1.91·11-s + 0.531·12-s + 0.174·13-s − 0.588·14-s − 0.299·15-s − 0.179·16-s + 0.961·17-s + 0.00893·18-s + 0.137·19-s − 0.161·20-s − 0.857·21-s + 1.30·22-s + 1.27·23-s − 1.03·24-s − 0.908·25-s − 0.119·26-s + 1.00·27-s − 0.461·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 43 | \( 1 \) |
| good | 2 | \( 1 + 1.92T + 8T^{2} \) |
| 3 | \( 1 + 5.16T + 27T^{2} \) |
| 5 | \( 1 - 3.37T + 125T^{2} \) |
| 7 | \( 1 - 15.9T + 343T^{2} \) |
| 11 | \( 1 + 69.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 8.18T + 2.19e3T^{2} \) |
| 17 | \( 1 - 67.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 11.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 140.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 166.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 27.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 32.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + 225.T + 6.89e4T^{2} \) |
| 47 | \( 1 + 203.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 15.9T + 1.48e5T^{2} \) |
| 59 | \( 1 - 453.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 19.9T + 2.26e5T^{2} \) |
| 67 | \( 1 - 402.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 876.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 963.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 986.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.11e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 11.0T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.18e3T + 9.12e5T^{2} \) |
| 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
−8.315381336898678946580985514110, −7.928861455063241607188761388129, −7.12125907358701523230923356670, −5.80513107184617750756100462602, −5.19130371021012672565750159152, −4.86670861446336096758617903055, −3.41786485284234159047067129268, −2.07237912359231556674535793458, −0.924539138919831740635414764955, 0,
0.924539138919831740635414764955, 2.07237912359231556674535793458, 3.41786485284234159047067129268, 4.86670861446336096758617903055, 5.19130371021012672565750159152, 5.80513107184617750756100462602, 7.12125907358701523230923356670, 7.928861455063241607188761388129, 8.315381336898678946580985514110
