L(s) = 1 | + 4·11-s − 8·23-s − 5·25-s + 2·29-s + 6·37-s + 12·43-s − 10·53-s − 4·67-s − 16·71-s + 8·79-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | + 1.20·11-s − 1.66·23-s − 25-s + 0.371·29-s + 0.986·37-s + 1.82·43-s − 1.37·53-s − 0.488·67-s − 1.89·71-s + 0.900·79-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−15.54807293408806, −14.77884831371177, −14.47422798244651, −13.88667526414848, −13.50383505144056, −12.76361498433701, −12.04642761098775, −11.95806254908191, −11.21398967826502, −10.67298368001286, −9.997015570092439, −9.430383977690555, −9.136609498471160, −8.241465489456920, −7.860820698644434, −7.222809916218860, −6.437271322318744, −6.050671549307894, −5.518750446294870, −4.395791086145914, −4.212103789657070, −3.461760767959582, −2.603063468016341, −1.845305487402979, −1.108801751878753, 0,
1.108801751878753, 1.845305487402979, 2.603063468016341, 3.461760767959582, 4.212103789657070, 4.395791086145914, 5.518750446294870, 6.050671549307894, 6.437271322318744, 7.222809916218860, 7.860820698644434, 8.241465489456920, 9.136609498471160, 9.430383977690555, 9.997015570092439, 10.67298368001286, 11.21398967826502, 11.95806254908191, 12.04642761098775, 12.76361498433701, 13.50383505144056, 13.88667526414848, 14.47422798244651, 14.77884831371177, 15.54807293408806