| L(s) = 1 | − 2-s − 2·3-s − 4-s − 4·5-s + 2·6-s − 3·7-s + 3·8-s + 9-s + 4·10-s − 4·11-s + 2·12-s − 5·13-s + 3·14-s + 8·15-s − 16-s − 3·17-s − 18-s − 4·19-s + 4·20-s + 6·21-s + 4·22-s + 8·23-s − 6·24-s + 11·25-s + 5·26-s + 4·27-s + 3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 1/2·4-s − 1.78·5-s + 0.816·6-s − 1.13·7-s + 1.06·8-s + 1/3·9-s + 1.26·10-s − 1.20·11-s + 0.577·12-s − 1.38·13-s + 0.801·14-s + 2.06·15-s − 1/4·16-s − 0.727·17-s − 0.235·18-s − 0.917·19-s + 0.894·20-s + 1.30·21-s + 0.852·22-s + 1.66·23-s − 1.22·24-s + 11/5·25-s + 0.980·26-s + 0.769·27-s + 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 433 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 433 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 433 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.65914823427479669007317594203, −9.307903179910026089353913014640, −8.432605385178941132054991521013, −7.41239186062132188132348593599, −6.81362847351019760597193961457, −5.16976335052993119549954852181, −4.53754586413642223166086062553, −3.13653467946233451337264151475, 0, 0,
3.13653467946233451337264151475, 4.53754586413642223166086062553, 5.16976335052993119549954852181, 6.81362847351019760597193961457, 7.41239186062132188132348593599, 8.432605385178941132054991521013, 9.307903179910026089353913014640, 10.65914823427479669007317594203
