L(s) = 1 | + 0.265·2-s − 0.631·3-s − 1.92·4-s + 5-s − 0.167·6-s + 3.82·7-s − 1.04·8-s − 2.60·9-s + 0.265·10-s + 6.10·11-s + 1.21·12-s + 2.37·13-s + 1.01·14-s − 0.631·15-s + 3.58·16-s − 4.90·17-s − 0.691·18-s − 1.03·19-s − 1.92·20-s − 2.41·21-s + 1.62·22-s + 1.57·23-s + 0.659·24-s + 25-s + 0.630·26-s + 3.53·27-s − 7.37·28-s + ⋯ |
L(s) = 1 | + 0.187·2-s − 0.364·3-s − 0.964·4-s + 0.447·5-s − 0.0685·6-s + 1.44·7-s − 0.369·8-s − 0.867·9-s + 0.0840·10-s + 1.83·11-s + 0.351·12-s + 0.657·13-s + 0.271·14-s − 0.163·15-s + 0.895·16-s − 1.18·17-s − 0.162·18-s − 0.237·19-s − 0.431·20-s − 0.526·21-s + 0.345·22-s + 0.329·23-s + 0.134·24-s + 0.200·25-s + 0.123·26-s + 0.680·27-s − 1.39·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.769521711\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.769521711\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
good | 2 | \( 1 - 0.265T + 2T^{2} \) |
| 3 | \( 1 + 0.631T + 3T^{2} \) |
| 7 | \( 1 - 3.82T + 7T^{2} \) |
| 11 | \( 1 - 6.10T + 11T^{2} \) |
| 13 | \( 1 - 2.37T + 13T^{2} \) |
| 17 | \( 1 + 4.90T + 17T^{2} \) |
| 19 | \( 1 + 1.03T + 19T^{2} \) |
| 23 | \( 1 - 1.57T + 23T^{2} \) |
| 29 | \( 1 + 3.97T + 29T^{2} \) |
| 31 | \( 1 + 4.98T + 31T^{2} \) |
| 37 | \( 1 - 11.9T + 37T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 43 | \( 1 + 8.00T + 43T^{2} \) |
| 47 | \( 1 + 12.7T + 47T^{2} \) |
| 53 | \( 1 - 8.99T + 53T^{2} \) |
| 59 | \( 1 - 3.97T + 59T^{2} \) |
| 61 | \( 1 + 9.14T + 61T^{2} \) |
| 67 | \( 1 - 1.34T + 67T^{2} \) |
| 71 | \( 1 - 12.8T + 71T^{2} \) |
| 73 | \( 1 + 1.78T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 - 2.64T + 83T^{2} \) |
| 89 | \( 1 - 6.23T + 89T^{2} \) |
| 97 | \( 1 + 0.219T + 97T^{2} \) |
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
−8.983617911136640246099548914291, −8.642057117865367523145627114454, −7.76869536612545656694790329000, −6.49229673399442670376454974124, −5.95471145076533411219243799631, −5.03074581069347317037520207979, −4.38918294050851876468253248328, −3.57642786134937509247354534831, −2.02929678398793282043256168985, −0.941520519228164545075769942337,
0.941520519228164545075769942337, 2.02929678398793282043256168985, 3.57642786134937509247354534831, 4.38918294050851876468253248328, 5.03074581069347317037520207979, 5.95471145076533411219243799631, 6.49229673399442670376454974124, 7.76869536612545656694790329000, 8.642057117865367523145627114454, 8.983617911136640246099548914291