| L(s) = 1 | + 2.11·2-s − 8.38·3-s − 3.51·4-s − 14.3·5-s − 17.7·6-s − 4.54·7-s − 24.3·8-s + 43.2·9-s − 30.4·10-s − 48.0·11-s + 29.4·12-s − 49.5·13-s − 9.63·14-s + 120.·15-s − 23.5·16-s + 91.6·18-s + 31.1·19-s + 50.4·20-s + 38.1·21-s − 101.·22-s − 41.8·23-s + 204.·24-s + 81.3·25-s − 104.·26-s − 136.·27-s + 15.9·28-s + 121.·29-s + ⋯ |
| L(s) = 1 | + 0.748·2-s − 1.61·3-s − 0.439·4-s − 1.28·5-s − 1.20·6-s − 0.245·7-s − 1.07·8-s + 1.60·9-s − 0.961·10-s − 1.31·11-s + 0.708·12-s − 1.05·13-s − 0.183·14-s + 2.07·15-s − 0.367·16-s + 1.20·18-s + 0.375·19-s + 0.564·20-s + 0.396·21-s − 0.987·22-s − 0.379·23-s + 1.73·24-s + 0.650·25-s − 0.791·26-s − 0.972·27-s + 0.107·28-s + 0.777·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2133340906\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2133340906\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 - 2.11T + 8T^{2} \) |
| 3 | \( 1 + 8.38T + 27T^{2} \) |
| 5 | \( 1 + 14.3T + 125T^{2} \) |
| 7 | \( 1 + 4.54T + 343T^{2} \) |
| 11 | \( 1 + 48.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 49.5T + 2.19e3T^{2} \) |
| 19 | \( 1 - 31.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 41.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 121.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 167.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 356.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 63.3T + 6.89e4T^{2} \) |
| 43 | \( 1 + 87.7T + 7.95e4T^{2} \) |
| 47 | \( 1 + 281.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 5.63T + 1.48e5T^{2} \) |
| 59 | \( 1 + 134.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 598.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 951.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 332.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 26.5T + 3.89e5T^{2} \) |
| 79 | \( 1 - 33.1T + 4.93e5T^{2} \) |
| 83 | \( 1 - 283.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 191.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 287.T + 9.12e5T^{2} \) |
| 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
−11.77729753868761285214835738682, −10.65552610278916636362183115208, −9.837424933639826408882132505142, −8.264627168516428461701653017577, −7.26632646687499383826149015211, −6.11441651341480121906859931701, −5.01960518568506237406478074322, −4.59478652472218458747477467065, −3.17693739731368489413689279857, −0.29776067725907282650450366172,
0.29776067725907282650450366172, 3.17693739731368489413689279857, 4.59478652472218458747477467065, 5.01960518568506237406478074322, 6.11441651341480121906859931701, 7.26632646687499383826149015211, 8.264627168516428461701653017577, 9.837424933639826408882132505142, 10.65552610278916636362183115208, 11.77729753868761285214835738682
