L(s) = 1 | + 2-s − 3-s + 4-s − 3·5-s − 6-s + 8-s + 9-s − 3·10-s + 2·11-s − 12-s + 3·15-s + 16-s + 17-s + 18-s − 3·19-s − 3·20-s + 2·22-s − 5·23-s − 24-s + 4·25-s − 27-s + 6·29-s + 3·30-s + 32-s − 2·33-s + 34-s + 36-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.34·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.948·10-s + 0.603·11-s − 0.288·12-s + 0.774·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 0.688·19-s − 0.670·20-s + 0.426·22-s − 1.04·23-s − 0.204·24-s + 4/5·25-s − 0.192·27-s + 1.11·29-s + 0.547·30-s + 0.176·32-s − 0.348·33-s + 0.171·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4998 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4998 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 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
−7.999584304806817648883741608179, −6.90085781732912113612720488363, −6.52917345311555964462663430705, −5.65978917732653846563824038203, −4.78281448685172331829472381036, −4.12895883242259426249446717112, −3.65753137916662577834804656291, −2.58347863740790163494421300998, −1.30426982546539298052700650903, 0,
1.30426982546539298052700650903, 2.58347863740790163494421300998, 3.65753137916662577834804656291, 4.12895883242259426249446717112, 4.78281448685172331829472381036, 5.65978917732653846563824038203, 6.52917345311555964462663430705, 6.90085781732912113612720488363, 7.999584304806817648883741608179