L(s) = 1 | + 3-s − 3·5-s − 2·7-s + 9-s + 5·11-s − 13-s − 3·15-s + 5·19-s − 2·21-s + 23-s + 4·25-s + 27-s − 6·29-s − 10·31-s + 5·33-s + 6·35-s − 2·37-s − 39-s − 5·41-s − 43-s − 3·45-s + 2·47-s − 3·49-s + 6·53-s − 15·55-s + 5·57-s + 10·61-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.34·5-s − 0.755·7-s + 1/3·9-s + 1.50·11-s − 0.277·13-s − 0.774·15-s + 1.14·19-s − 0.436·21-s + 0.208·23-s + 4/5·25-s + 0.192·27-s − 1.11·29-s − 1.79·31-s + 0.870·33-s + 1.01·35-s − 0.328·37-s − 0.160·39-s − 0.780·41-s − 0.152·43-s − 0.447·45-s + 0.291·47-s − 3/7·49-s + 0.824·53-s − 2.02·55-s + 0.662·57-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−16.38947332523206, −15.82295706474725, −15.37837560853260, −14.61039742715755, −14.51134311792962, −13.65911300931507, −13.03502051660333, −12.45128578555656, −11.93590430859236, −11.42054781177861, −10.94710425445737, −9.929784163747684, −9.476593427588277, −8.991683801144023, −8.350509445669262, −7.648094134738322, −7.043401942036943, −6.794131324118863, −5.727702575480353, −5.008830281177402, −3.958072902179980, −3.706663877390558, −3.223165766621572, −2.096566943307238, −1.099046017988340, 0,
1.099046017988340, 2.096566943307238, 3.223165766621572, 3.706663877390558, 3.958072902179980, 5.008830281177402, 5.727702575480353, 6.794131324118863, 7.043401942036943, 7.648094134738322, 8.350509445669262, 8.991683801144023, 9.476593427588277, 9.929784163747684, 10.94710425445737, 11.42054781177861, 11.93590430859236, 12.45128578555656, 13.03502051660333, 13.65911300931507, 14.51134311792962, 14.61039742715755, 15.37837560853260, 15.82295706474725, 16.38947332523206