L(s) = 1 | + (0.5 + 1.65i)3-s − 2·4-s − 3.31i·5-s + (−2.5 + 1.65i)9-s + 3.31i·11-s + (−1 − 3.31i)12-s + (5.5 − 1.65i)15-s + 4·16-s + 6.63i·20-s − 3.31i·23-s − 6·25-s + (−4 − 3.31i)27-s + 5·31-s + (−5.5 + 1.65i)33-s + (5 − 3.31i)36-s − 7·37-s + ⋯ |
L(s) = 1 | + (0.288 + 0.957i)3-s − 4-s − 1.48i·5-s + (−0.833 + 0.552i)9-s + 1.00i·11-s + (−0.288 − 0.957i)12-s + (1.42 − 0.428i)15-s + 16-s + 1.48i·20-s − 0.691i·23-s − 1.20·25-s + (−0.769 − 0.638i)27-s + 0.898·31-s + (−0.957 + 0.288i)33-s + (0.833 − 0.552i)36-s − 1.15·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.660751 + 0.0974455i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.660751 + 0.0974455i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.5 - 1.65i)T \) |
| 11 | \( 1 - 3.31iT \) |
good | 2 | \( 1 + 2T^{2} \) |
| 5 | \( 1 + 3.31iT - 5T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 3.31iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 + 7T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 6.63iT - 47T^{2} \) |
| 53 | \( 1 + 13.2iT - 53T^{2} \) |
| 59 | \( 1 + 3.31iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 13T + 67T^{2} \) |
| 71 | \( 1 - 16.5iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 16.5iT - 89T^{2} \) |
| 97 | \( 1 - 17T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.85574838471113963425433979456, −15.74352154365985779192189445348, −14.48010355709446529915257600066, −13.25737799010645723889294192930, −12.18377057708802742812991424989, −10.14183943032400540896029256073, −9.160132383755029723980325860329, −8.264947596125555144370168258865, −5.18130267836277930636751198273, −4.27630887656717266868263341494,
3.25699087747366984275342802206, 6.01987084005356094215710711834, 7.48410396012403277235033293890, 8.841889407311114617556569512502, 10.50315208485772406638675869216, 11.91285142562008747901103505623, 13.52133580755889374274588661744, 14.04167765341748268011552322659, 15.15355040523506862802895652520, 17.17113961041653810289302193238