L(s) = 1 | + 3·5-s + 7-s + 3·11-s − 4·13-s − 2·19-s + 6·23-s + 4·25-s + 6·29-s − 5·31-s + 3·35-s + 2·37-s − 6·41-s + 10·43-s − 6·47-s − 6·49-s + 9·53-s + 9·55-s − 12·59-s + 8·61-s − 12·65-s − 14·67-s − 7·73-s + 3·77-s − 8·79-s + 3·83-s − 18·89-s − 4·91-s + ⋯ |
L(s) = 1 | + 1.34·5-s + 0.377·7-s + 0.904·11-s − 1.10·13-s − 0.458·19-s + 1.25·23-s + 4/5·25-s + 1.11·29-s − 0.898·31-s + 0.507·35-s + 0.328·37-s − 0.937·41-s + 1.52·43-s − 0.875·47-s − 6/7·49-s + 1.23·53-s + 1.21·55-s − 1.56·59-s + 1.02·61-s − 1.48·65-s − 1.71·67-s − 0.819·73-s + 0.341·77-s − 0.900·79-s + 0.329·83-s − 1.90·89-s − 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.784916201\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.784916201\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−11.06563679257021720417991599271, −10.13919133269169682063036547080, −9.405159791372798554846122664326, −8.657482823158305382635077732253, −7.29929175763847648434390847974, −6.43760550649778711412630154056, −5.43287815412844995352588032205, −4.47356450228848094997188248522, −2.78595567768432254175608596610, −1.56286118008502235583804201463,
1.56286118008502235583804201463, 2.78595567768432254175608596610, 4.47356450228848094997188248522, 5.43287815412844995352588032205, 6.43760550649778711412630154056, 7.29929175763847648434390847974, 8.657482823158305382635077732253, 9.405159791372798554846122664326, 10.13919133269169682063036547080, 11.06563679257021720417991599271