L(s) = 1 | − 2-s − 3-s − 4-s − 4·5-s + 6-s − 7-s + 3·8-s − 2·9-s + 4·10-s − 6·11-s + 12-s − 5·13-s + 14-s + 4·15-s − 16-s − 7·17-s + 2·18-s − 4·19-s + 4·20-s + 21-s + 6·22-s − 4·23-s − 3·24-s + 11·25-s + 5·26-s + 5·27-s + 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s − 1.78·5-s + 0.408·6-s − 0.377·7-s + 1.06·8-s − 2/3·9-s + 1.26·10-s − 1.80·11-s + 0.288·12-s − 1.38·13-s + 0.267·14-s + 1.03·15-s − 1/4·16-s − 1.69·17-s + 0.471·18-s − 0.917·19-s + 0.894·20-s + 0.218·21-s + 1.27·22-s − 0.834·23-s − 0.612·24-s + 11/5·25-s + 0.980·26-s + 0.962·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18097 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18097 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 18097 | \( 1 + T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−16.66355078553868, −16.12723187140413, −15.66465705717195, −15.08798951136561, −14.69033501166069, −13.80715341856277, −13.12891148169160, −12.62186029198830, −12.31386071185314, −11.31674063975217, −11.12474175457839, −10.59241454938144, −9.972604400204676, −9.235211865855686, −8.578833682213959, −8.093201974911126, −7.703540173776869, −7.120167421868357, −6.410646553189068, −5.409430294898282, −4.732797424415094, −4.536545453925919, −3.577472295292634, −2.801843415547256, −1.946680073210762, 0, 0, 0,
1.946680073210762, 2.801843415547256, 3.577472295292634, 4.536545453925919, 4.732797424415094, 5.409430294898282, 6.410646553189068, 7.120167421868357, 7.703540173776869, 8.093201974911126, 8.578833682213959, 9.235211865855686, 9.972604400204676, 10.59241454938144, 11.12474175457839, 11.31674063975217, 12.31386071185314, 12.62186029198830, 13.12891148169160, 13.80715341856277, 14.69033501166069, 15.08798951136561, 15.66465705717195, 16.12723187140413, 16.66355078553868