L(s) = 1 | − 2-s − 2·3-s − 4-s + 2·6-s + 7-s + 3·8-s + 9-s + 3·11-s + 2·12-s + 3·13-s − 14-s − 16-s − 5·17-s − 18-s − 7·19-s − 2·21-s − 3·22-s − 3·23-s − 6·24-s − 5·25-s − 3·26-s + 4·27-s − 28-s + 7·31-s − 5·32-s − 6·33-s + 5·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.15·3-s − 1/2·4-s + 0.816·6-s + 0.377·7-s + 1.06·8-s + 1/3·9-s + 0.904·11-s + 0.577·12-s + 0.832·13-s − 0.267·14-s − 1/4·16-s − 1.21·17-s − 0.235·18-s − 1.60·19-s − 0.436·21-s − 0.639·22-s − 0.625·23-s − 1.22·24-s − 25-s − 0.588·26-s + 0.769·27-s − 0.188·28-s + 1.25·31-s − 0.883·32-s − 1.04·33-s + 0.857·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 443 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 443 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 443 | \( 1 + T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−10.69819795490158406782422490462, −9.881781756372744287873890056581, −8.701661060815549683587536680743, −8.285292308235240080848380600808, −6.74959768754122577112491189067, −6.09630380700586307112961369058, −4.80628065334994799877239926769, −4.03480921686296141905054645465, −1.65024885908346640526056366423, 0,
1.65024885908346640526056366423, 4.03480921686296141905054645465, 4.80628065334994799877239926769, 6.09630380700586307112961369058, 6.74959768754122577112491189067, 8.285292308235240080848380600808, 8.701661060815549683587536680743, 9.881781756372744287873890056581, 10.69819795490158406782422490462