| L(s) = 1 | − 2-s − 2·3-s − 4-s + 2·6-s − 2·7-s + 3·8-s + 9-s − 2·11-s + 2·12-s − 2·13-s + 2·14-s − 16-s − 3·17-s − 18-s − 2·19-s + 4·21-s + 2·22-s + 3·23-s − 6·24-s − 5·25-s + 2·26-s + 4·27-s + 2·28-s − 3·31-s − 5·32-s + 4·33-s + 3·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 1/2·4-s + 0.816·6-s − 0.755·7-s + 1.06·8-s + 1/3·9-s − 0.603·11-s + 0.577·12-s − 0.554·13-s + 0.534·14-s − 1/4·16-s − 0.727·17-s − 0.235·18-s − 0.458·19-s + 0.872·21-s + 0.426·22-s + 0.625·23-s − 1.22·24-s − 25-s + 0.392·26-s + 0.769·27-s + 0.377·28-s − 0.538·31-s − 0.883·32-s + 0.696·33-s + 0.514·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 6043 | \( 1+O(T) \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−7.36139491900651933737193241064, −6.66207825310479573386762668834, −5.99062580561940198415694539198, −5.16265582785539618666271884673, −4.70216219230077914572140251451, −3.75754787973003209894863901188, −2.67362633139362559265362765052, −1.48848649023931366618662242293, 0, 0,
1.48848649023931366618662242293, 2.67362633139362559265362765052, 3.75754787973003209894863901188, 4.70216219230077914572140251451, 5.16265582785539618666271884673, 5.99062580561940198415694539198, 6.66207825310479573386762668834, 7.36139491900651933737193241064
