L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 8-s + 9-s − 10-s − 3·11-s + 12-s − 2·13-s − 15-s + 16-s − 17-s + 18-s − 6·19-s − 20-s − 3·22-s + 2·23-s + 24-s − 4·25-s − 2·26-s + 27-s + 5·29-s − 30-s + 3·31-s + 32-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 + 1/3·9-s − 0.316·10-s − 0.904·11-s + 0.288·12-s − 0.554·13-s − 0.258·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 1.37·19-s − 0.223·20-s − 0.639·22-s + 0.417·23-s + 0.204·24-s − 4/5·25-s − 0.392·26-s + 0.192·27-s + 0.928·29-s − 0.182·30-s + 0.538·31-s + 0.176·32-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 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 13 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−7.83769224626347840665222148542, −7.21089255349474679578992197761, −6.44482390468926811562042875132, −5.65075075347149232741143780899, −4.64165647791889282087941350471, −4.32660396303858718077506510116, −3.22324571032770475332496321766, −2.62877423120311403316357506336, −1.71289844637639800614846436708, 0,
1.71289844637639800614846436708, 2.62877423120311403316357506336, 3.22324571032770475332496321766, 4.32660396303858718077506510116, 4.64165647791889282087941350471, 5.65075075347149232741143780899, 6.44482390468926811562042875132, 7.21089255349474679578992197761, 7.83769224626347840665222148542