L(s) = 1 | − 3-s − 5-s + 9-s − 11-s − 13-s + 15-s − 17-s − 19-s + 5·23-s − 4·25-s − 27-s − 2·29-s − 2·31-s + 33-s + 8·37-s + 39-s − 5·41-s − 5·43-s − 45-s − 12·47-s + 51-s + 55-s + 57-s + 2·59-s + 2·61-s + 65-s − 4·67-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.301·11-s − 0.277·13-s + 0.258·15-s − 0.242·17-s − 0.229·19-s + 1.04·23-s − 4/5·25-s − 0.192·27-s − 0.371·29-s − 0.359·31-s + 0.174·33-s + 1.31·37-s + 0.160·39-s − 0.780·41-s − 0.762·43-s − 0.149·45-s − 1.75·47-s + 0.140·51-s + 0.134·55-s + 0.132·57-s + 0.260·59-s + 0.256·61-s + 0.124·65-s − 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−13.33723890291711, −13.01811641767198, −12.62329738655532, −11.99249386459425, −11.54777480900443, −11.24527173204127, −10.77310048747070, −10.21208667485851, −9.697098448775391, −9.339426703957879, −8.642452125597169, −8.136669245506554, −7.700444703724849, −7.186961511756313, −6.608802352136144, −6.275526222762634, −5.511498041733336, −5.110697946026066, −4.623880416821329, −4.030526600278771, −3.455329712796425, −2.874515817636044, −2.115671399588663, −1.529501487538492, −0.6570868568538273, 0,
0.6570868568538273, 1.529501487538492, 2.115671399588663, 2.874515817636044, 3.455329712796425, 4.030526600278771, 4.623880416821329, 5.110697946026066, 5.511498041733336, 6.275526222762634, 6.608802352136144, 7.186961511756313, 7.700444703724849, 8.136669245506554, 8.642452125597169, 9.339426703957879, 9.697098448775391, 10.21208667485851, 10.77310048747070, 11.24527173204127, 11.54777480900443, 11.99249386459425, 12.62329738655532, 13.01811641767198, 13.33723890291711