| L(s) = 1 | − 2-s − 3-s + 4-s − 1.69·5-s + 6-s + 7-s − 8-s + 9-s + 1.69·10-s + 2.60·11-s − 12-s − 13-s − 14-s + 1.69·15-s + 16-s − 17-s − 18-s + 1.24·19-s − 1.69·20-s − 21-s − 2.60·22-s − 4.89·23-s + 24-s − 2.13·25-s + 26-s − 27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.756·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.535·10-s + 0.785·11-s − 0.288·12-s − 0.277·13-s − 0.267·14-s + 0.436·15-s + 0.250·16-s − 0.242·17-s − 0.235·18-s + 0.286·19-s − 0.378·20-s − 0.218·21-s − 0.555·22-s − 1.02·23-s + 0.204·24-s − 0.427·25-s + 0.196·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9282 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9282 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7698474786\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7698474786\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 1.69T + 5T^{2} \) |
| 11 | \( 1 - 2.60T + 11T^{2} \) |
| 19 | \( 1 - 1.24T + 19T^{2} \) |
| 23 | \( 1 + 4.89T + 23T^{2} \) |
| 29 | \( 1 - 6.29T + 29T^{2} \) |
| 31 | \( 1 + 8.63T + 31T^{2} \) |
| 37 | \( 1 + 6.85T + 37T^{2} \) |
| 41 | \( 1 + 7.00T + 41T^{2} \) |
| 43 | \( 1 - 2.26T + 43T^{2} \) |
| 47 | \( 1 - 12.0T + 47T^{2} \) |
| 53 | \( 1 - 7.25T + 53T^{2} \) |
| 59 | \( 1 - 1.45T + 59T^{2} \) |
| 61 | \( 1 - 7.62T + 61T^{2} \) |
| 67 | \( 1 - 7.83T + 67T^{2} \) |
| 71 | \( 1 + 8.39T + 71T^{2} \) |
| 73 | \( 1 - 2.02T + 73T^{2} \) |
| 79 | \( 1 + 7.40T + 79T^{2} \) |
| 83 | \( 1 - 3.06T + 83T^{2} \) |
| 89 | \( 1 - 5.67T + 89T^{2} \) |
| 97 | \( 1 + 1.91T + 97T^{2} \) |
| 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
−7.59662676159935663678022707367, −7.18993499718956740154935204310, −6.51174826473029923199348554728, −5.72355807338991134627038359179, −5.04103903789489447214342008912, −4.06503122846104606831830121512, −3.64730137946482113248331401707, −2.37906643135086861562512884819, −1.53517742638697454885116992421, −0.49871736264633848623295675104,
0.49871736264633848623295675104, 1.53517742638697454885116992421, 2.37906643135086861562512884819, 3.64730137946482113248331401707, 4.06503122846104606831830121512, 5.04103903789489447214342008912, 5.72355807338991134627038359179, 6.51174826473029923199348554728, 7.18993499718956740154935204310, 7.59662676159935663678022707367
