| L(s) = 1 | − 2-s − 2·3-s − 4-s + 5-s + 2·6-s − 7-s + 3·8-s + 9-s − 10-s − 4·11-s + 2·12-s + 5·13-s + 14-s − 2·15-s − 16-s − 3·17-s − 18-s + 19-s − 20-s + 2·21-s + 4·22-s − 23-s − 6·24-s + 25-s − 5·26-s + 4·27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 1/2·4-s + 0.447·5-s + 0.816·6-s − 0.377·7-s + 1.06·8-s + 1/3·9-s − 0.316·10-s − 1.20·11-s + 0.577·12-s + 1.38·13-s + 0.267·14-s − 0.516·15-s − 1/4·16-s − 0.727·17-s − 0.235·18-s + 0.229·19-s − 0.223·20-s + 0.436·21-s + 0.852·22-s − 0.208·23-s − 1.22·24-s + 1/5·25-s − 0.980·26-s + 0.769·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33635 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33635 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
| 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
−15.66530513251669, −14.72139317061383, −13.99898555018789, −13.62481681130328, −13.05617506983195, −12.77620112372150, −12.04872362947945, −11.28073090424630, −10.91289715683334, −10.49381718615917, −10.04221394163116, −9.381070979723195, −8.850426634060980, −8.273843321459027, −7.833925922544277, −6.931832937202195, −6.498603042526916, −5.639966307437083, −5.579514941270811, −4.669995721992515, −4.223217357108101, −3.265512374355086, −2.481720024427142, −1.488036539766400, −0.7522922231632420, 0,
0.7522922231632420, 1.488036539766400, 2.481720024427142, 3.265512374355086, 4.223217357108101, 4.669995721992515, 5.579514941270811, 5.639966307437083, 6.498603042526916, 6.931832937202195, 7.833925922544277, 8.273843321459027, 8.850426634060980, 9.381070979723195, 10.04221394163116, 10.49381718615917, 10.91289715683334, 11.28073090424630, 12.04872362947945, 12.77620112372150, 13.05617506983195, 13.62481681130328, 13.99898555018789, 14.72139317061383, 15.66530513251669
