L(s) = 1 | + 3-s − 2·7-s + 9-s − 4·11-s − 2·13-s − 3·17-s − 2·21-s + 23-s + 27-s + 6·29-s − 7·31-s − 4·33-s − 2·37-s − 2·39-s + 5·41-s − 43-s − 7·47-s − 3·49-s − 3·51-s + 5·53-s + 5·59-s + 8·61-s − 2·63-s − 9·67-s + 69-s − 4·71-s − 8·73-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.727·17-s − 0.436·21-s + 0.208·23-s + 0.192·27-s + 1.11·29-s − 1.25·31-s − 0.696·33-s − 0.328·37-s − 0.320·39-s + 0.780·41-s − 0.152·43-s − 1.02·47-s − 3/7·49-s − 0.420·51-s + 0.686·53-s + 0.650·59-s + 1.02·61-s − 0.251·63-s − 1.09·67-s + 0.120·69-s − 0.474·71-s − 0.936·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 43 | \( 1 + T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−13.25132315342373, −12.95151801194127, −12.44031323504260, −12.00090876359705, −11.29556580454852, −10.92675313182108, −10.24374273867201, −10.06855973110353, −9.534715486685225, −8.973832874171661, −8.567746549980500, −8.103650050076161, −7.489998381361843, −7.107650638982588, −6.685660523241354, −6.009093027932523, −5.511399636862789, −4.918121892513517, −4.457394823057294, −3.851103970844312, −3.098644165535900, −2.879450898734532, −2.222048897248905, −1.693035158807152, −0.6642531581334232, 0,
0.6642531581334232, 1.693035158807152, 2.222048897248905, 2.879450898734532, 3.098644165535900, 3.851103970844312, 4.457394823057294, 4.918121892513517, 5.511399636862789, 6.009093027932523, 6.685660523241354, 7.107650638982588, 7.489998381361843, 8.103650050076161, 8.567746549980500, 8.973832874171661, 9.534715486685225, 10.06855973110353, 10.24374273867201, 10.92675313182108, 11.29556580454852, 12.00090876359705, 12.44031323504260, 12.95151801194127, 13.25132315342373