L(s) = 1 | + 2-s − 1.34·3-s + 4-s − 5-s − 1.34·6-s + 0.629·7-s + 8-s − 1.20·9-s − 10-s + 5.31·11-s − 1.34·12-s − 2.28·13-s + 0.629·14-s + 1.34·15-s + 16-s − 5.32·17-s − 1.20·18-s + 3.82·19-s − 20-s − 0.843·21-s + 5.31·22-s + 2.83·23-s − 1.34·24-s + 25-s − 2.28·26-s + 5.63·27-s + 0.629·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.773·3-s + 0.5·4-s − 0.447·5-s − 0.547·6-s + 0.237·7-s + 0.353·8-s − 0.401·9-s − 0.316·10-s + 1.60·11-s − 0.386·12-s − 0.632·13-s + 0.168·14-s + 0.346·15-s + 0.250·16-s − 1.29·17-s − 0.283·18-s + 0.878·19-s − 0.223·20-s − 0.184·21-s + 1.13·22-s + 0.590·23-s − 0.273·24-s + 0.200·25-s − 0.447·26-s + 1.08·27-s + 0.118·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.057904320\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.057904320\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
good | 3 | \( 1 + 1.34T + 3T^{2} \) |
| 7 | \( 1 - 0.629T + 7T^{2} \) |
| 11 | \( 1 - 5.31T + 11T^{2} \) |
| 13 | \( 1 + 2.28T + 13T^{2} \) |
| 17 | \( 1 + 5.32T + 17T^{2} \) |
| 19 | \( 1 - 3.82T + 19T^{2} \) |
| 23 | \( 1 - 2.83T + 23T^{2} \) |
| 29 | \( 1 - 6.56T + 29T^{2} \) |
| 31 | \( 1 + 9.22T + 31T^{2} \) |
| 37 | \( 1 + 1.66T + 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 - 7.74T + 43T^{2} \) |
| 47 | \( 1 - 6.08T + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 - 2.23T + 59T^{2} \) |
| 61 | \( 1 + 1.18T + 61T^{2} \) |
| 67 | \( 1 - 13.2T + 67T^{2} \) |
| 71 | \( 1 - 4.77T + 71T^{2} \) |
| 73 | \( 1 + 7.38T + 73T^{2} \) |
| 79 | \( 1 + 8.69T + 79T^{2} \) |
| 83 | \( 1 - 8.56T + 83T^{2} \) |
| 89 | \( 1 - 3.05T + 89T^{2} \) |
| 97 | \( 1 + 4.45T + 97T^{2} \) |
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
−8.578801406966633914026884081426, −7.33341815150596262690037830763, −6.87865450021153805764061592748, −6.22732505216206635255190034476, −5.34769717287797988175321811240, −4.76355344702669157883943313773, −3.98197378877533731000350819198, −3.16085380919432150127673197862, −2.02337803597749100464969163860, −0.77180540327825472432279612816,
0.77180540327825472432279612816, 2.02337803597749100464969163860, 3.16085380919432150127673197862, 3.98197378877533731000350819198, 4.76355344702669157883943313773, 5.34769717287797988175321811240, 6.22732505216206635255190034476, 6.87865450021153805764061592748, 7.33341815150596262690037830763, 8.578801406966633914026884081426