L(s) = 1 | + 3·3-s − 7-s + 6·9-s − 6·13-s + 7·17-s − 5·19-s − 3·21-s − 5·25-s + 9·27-s − 8·29-s − 2·31-s − 4·37-s − 18·39-s + 6·41-s + 3·43-s + 8·47-s + 49-s + 21·51-s − 6·53-s − 15·57-s − 5·59-s − 10·61-s − 6·63-s − 13·67-s − 6·71-s − 11·73-s − 15·75-s + ⋯ |
L(s) = 1 | + 1.73·3-s − 0.377·7-s + 2·9-s − 1.66·13-s + 1.69·17-s − 1.14·19-s − 0.654·21-s − 25-s + 1.73·27-s − 1.48·29-s − 0.359·31-s − 0.657·37-s − 2.88·39-s + 0.937·41-s + 0.457·43-s + 1.16·47-s + 1/7·49-s + 2.94·51-s − 0.824·53-s − 1.98·57-s − 0.650·59-s − 1.28·61-s − 0.755·63-s − 1.58·67-s − 0.712·71-s − 1.28·73-s − 1.73·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.589207754\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.589207754\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−13.01215872757019, −12.44509195032040, −12.24372584032399, −11.69493692148183, −10.82417745134815, −10.43758275408201, −9.835395023424701, −9.692689190756011, −9.087097300871634, −8.847831844331187, −8.144452167230367, −7.552777119036624, −7.456458501332791, −7.162265460656909, −6.056179144766165, −5.872205578482116, −5.088553431140461, −4.393647821519365, −4.065068498762424, −3.435812684018798, −2.944791090719153, −2.558148423478045, −1.842092704399824, −1.543505675430307, −0.3556852453147676,
0.3556852453147676, 1.543505675430307, 1.842092704399824, 2.558148423478045, 2.944791090719153, 3.435812684018798, 4.065068498762424, 4.393647821519365, 5.088553431140461, 5.872205578482116, 6.056179144766165, 7.162265460656909, 7.456458501332791, 7.552777119036624, 8.144452167230367, 8.847831844331187, 9.087097300871634, 9.692689190756011, 9.835395023424701, 10.43758275408201, 10.82417745134815, 11.69493692148183, 12.24372584032399, 12.44509195032040, 13.01215872757019