L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 8-s − 2·9-s − 10-s + 11-s + 12-s + 2·13-s + 15-s + 16-s + 2·18-s − 3·19-s + 20-s − 22-s − 4·23-s − 24-s − 4·25-s − 2·26-s − 5·27-s + 8·29-s − 30-s − 10·31-s − 32-s + 33-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.353·8-s − 2/3·9-s − 0.316·10-s + 0.301·11-s + 0.288·12-s + 0.554·13-s + 0.258·15-s + 1/4·16-s + 0.471·18-s − 0.688·19-s + 0.223·20-s − 0.213·22-s − 0.834·23-s − 0.204·24-s − 4/5·25-s − 0.392·26-s − 0.962·27-s + 1.48·29-s − 0.182·30-s − 1.79·31-s − 0.176·32-s + 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−8.235488157752256092850697786452, −7.60089326266437508387406555200, −6.57926114029465742516465727308, −6.07205202553666667646205912274, −5.25416105194204324830065785895, −4.03462785696889550178959558360, −3.26074243368339578405759497999, −2.29287365196976319602323178222, −1.55785935167123907674885589319, 0,
1.55785935167123907674885589319, 2.29287365196976319602323178222, 3.26074243368339578405759497999, 4.03462785696889550178959558360, 5.25416105194204324830065785895, 6.07205202553666667646205912274, 6.57926114029465742516465727308, 7.60089326266437508387406555200, 8.235488157752256092850697786452