| L(s) = 1 | − 2-s − 3-s + 4-s − 0.651·5-s + 6-s + 2.08·7-s − 8-s + 9-s + 0.651·10-s + 2.36·11-s − 12-s − 2.71·13-s − 2.08·14-s + 0.651·15-s + 16-s + 2.65·17-s − 18-s − 2.65·19-s − 0.651·20-s − 2.08·21-s − 2.36·22-s − 23-s + 24-s − 4.57·25-s + 2.71·26-s − 27-s + 2.08·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.291·5-s + 0.408·6-s + 0.786·7-s − 0.353·8-s + 0.333·9-s + 0.206·10-s + 0.712·11-s − 0.288·12-s − 0.751·13-s − 0.556·14-s + 0.168·15-s + 0.250·16-s + 0.644·17-s − 0.235·18-s − 0.610·19-s − 0.145·20-s − 0.454·21-s − 0.503·22-s − 0.208·23-s + 0.204·24-s − 0.915·25-s + 0.531·26-s − 0.192·27-s + 0.393·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 + 0.651T + 5T^{2} \) |
| 7 | \( 1 - 2.08T + 7T^{2} \) |
| 11 | \( 1 - 2.36T + 11T^{2} \) |
| 13 | \( 1 + 2.71T + 13T^{2} \) |
| 17 | \( 1 - 2.65T + 17T^{2} \) |
| 19 | \( 1 + 2.65T + 19T^{2} \) |
| 31 | \( 1 - 2.23T + 31T^{2} \) |
| 37 | \( 1 + 2.97T + 37T^{2} \) |
| 41 | \( 1 - 3.90T + 41T^{2} \) |
| 43 | \( 1 + 0.808T + 43T^{2} \) |
| 47 | \( 1 + 6.36T + 47T^{2} \) |
| 53 | \( 1 + 6.79T + 53T^{2} \) |
| 59 | \( 1 + 1.69T + 59T^{2} \) |
| 61 | \( 1 + 3.77T + 61T^{2} \) |
| 67 | \( 1 - 6.66T + 67T^{2} \) |
| 71 | \( 1 + 1.98T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 + 6.20T + 79T^{2} \) |
| 83 | \( 1 - 14.8T + 83T^{2} \) |
| 89 | \( 1 - 15.4T + 89T^{2} \) |
| 97 | \( 1 - 2.79T + 97T^{2} \) |
| 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
−7.933517750384275376642598125071, −7.58022635873971674715027440313, −6.65216112119638871328944094015, −6.01933248358598366932045494336, −5.08994522946566793733403671114, −4.37330929900252047456446448045, −3.41441025034132854064354956597, −2.14371433237439539368538134572, −1.30059204841024934312259414388, 0,
1.30059204841024934312259414388, 2.14371433237439539368538134572, 3.41441025034132854064354956597, 4.37330929900252047456446448045, 5.08994522946566793733403671114, 6.01933248358598366932045494336, 6.65216112119638871328944094015, 7.58022635873971674715027440313, 7.933517750384275376642598125071
