| L(s) = 1 | − 3·9-s + 13-s − 6·17-s − 4·19-s + 23-s − 5·25-s − 2·29-s − 10·31-s + 4·37-s − 10·41-s + 4·43-s + 6·47-s − 2·53-s + 2·59-s − 10·61-s + 16·67-s − 10·71-s + 2·73-s + 8·79-s + 9·81-s + 16·83-s + 8·89-s + 4·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 9-s + 0.277·13-s − 1.45·17-s − 0.917·19-s + 0.208·23-s − 25-s − 0.371·29-s − 1.79·31-s + 0.657·37-s − 1.56·41-s + 0.609·43-s + 0.875·47-s − 0.274·53-s + 0.260·59-s − 1.28·61-s + 1.95·67-s − 1.18·71-s + 0.234·73-s + 0.900·79-s + 81-s + 1.75·83-s + 0.847·89-s + 0.406·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 234416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−13.21863720107121, −12.75583330451870, −12.12656237383949, −11.78218949069534, −11.09349817278852, −10.96290099486733, −10.57494251645890, −9.828618234774585, −9.181098599481609, −9.045578861199797, −8.516695364010681, −7.979539733974537, −7.569930311736677, −6.829437944590068, −6.532053826626046, −5.870436114499893, −5.613762569595675, −4.877778739569011, −4.447382555938977, −3.663055136967486, −3.516871501944114, −2.536402518939033, −2.166787991659132, −1.668015581952756, −0.5785682734803539, 0,
0.5785682734803539, 1.668015581952756, 2.166787991659132, 2.536402518939033, 3.516871501944114, 3.663055136967486, 4.447382555938977, 4.877778739569011, 5.613762569595675, 5.870436114499893, 6.532053826626046, 6.829437944590068, 7.569930311736677, 7.979539733974537, 8.516695364010681, 9.045578861199797, 9.181098599481609, 9.828618234774585, 10.57494251645890, 10.96290099486733, 11.09349817278852, 11.78218949069534, 12.12656237383949, 12.75583330451870, 13.21863720107121
