L(s) = 1 | − 3-s + 2·5-s − 3·7-s − 2·9-s − 11-s − 13-s − 2·15-s + 4·19-s + 3·21-s − 3·23-s − 25-s + 5·27-s + 10·31-s + 33-s − 6·35-s + 4·37-s + 39-s − 2·41-s − 5·43-s − 4·45-s − 5·47-s + 2·49-s − 12·53-s − 2·55-s − 4·57-s + 4·59-s + 7·61-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.894·5-s − 1.13·7-s − 2/3·9-s − 0.301·11-s − 0.277·13-s − 0.516·15-s + 0.917·19-s + 0.654·21-s − 0.625·23-s − 1/5·25-s + 0.962·27-s + 1.79·31-s + 0.174·33-s − 1.01·35-s + 0.657·37-s + 0.160·39-s − 0.312·41-s − 0.762·43-s − 0.596·45-s − 0.729·47-s + 2/7·49-s − 1.64·53-s − 0.269·55-s − 0.529·57-s + 0.520·59-s + 0.896·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 + T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 5 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−15.56027723378047, −14.59890578886325, −14.24999553940518, −13.64525495404640, −13.22280777735610, −12.77339889553563, −11.99719734673659, −11.73459943075781, −11.11002106165521, −10.30948019196922, −9.953005786880938, −9.599985052573262, −9.009341976600121, −8.147756351002348, −7.820984353022746, −6.676101280932852, −6.526005229772612, −5.954792835827528, −5.331864924863492, −4.920880633146671, −3.971526978031289, −3.106682545564311, −2.742382731725251, −1.892978542324595, −0.8619742161090075, 0,
0.8619742161090075, 1.892978542324595, 2.742382731725251, 3.106682545564311, 3.971526978031289, 4.920880633146671, 5.331864924863492, 5.954792835827528, 6.526005229772612, 6.676101280932852, 7.820984353022746, 8.147756351002348, 9.009341976600121, 9.599985052573262, 9.953005786880938, 10.30948019196922, 11.11002106165521, 11.73459943075781, 11.99719734673659, 12.77339889553563, 13.22280777735610, 13.64525495404640, 14.24999553940518, 14.59890578886325, 15.56027723378047