| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s + 11-s − 3·13-s − 14-s + 16-s − 8·17-s − 3·19-s − 22-s + 6·23-s + 3·26-s + 28-s + 6·29-s − 4·31-s − 32-s + 8·34-s − 2·37-s + 3·38-s + 11·41-s − 43-s + 44-s − 6·46-s + 47-s + 49-s − 3·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s + 0.301·11-s − 0.832·13-s − 0.267·14-s + 1/4·16-s − 1.94·17-s − 0.688·19-s − 0.213·22-s + 1.25·23-s + 0.588·26-s + 0.188·28-s + 1.11·29-s − 0.718·31-s − 0.176·32-s + 1.37·34-s − 0.328·37-s + 0.486·38-s + 1.71·41-s − 0.152·43-s + 0.150·44-s − 0.884·46-s + 0.145·47-s + 1/7·49-s − 0.416·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 11 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.27880137168187846608299322939, −6.84969850768552941427292187969, −6.24214304046055983576804811868, −5.25187531798959533490935874437, −4.59362908984002508530607246141, −3.89587893219936893189156628901, −2.65370407929883538179625442560, −2.23386593789675195169135037331, −1.13283480423575320064729724308, 0,
1.13283480423575320064729724308, 2.23386593789675195169135037331, 2.65370407929883538179625442560, 3.89587893219936893189156628901, 4.59362908984002508530607246141, 5.25187531798959533490935874437, 6.24214304046055983576804811868, 6.84969850768552941427292187969, 7.27880137168187846608299322939
