L(s) = 1 | + (0.984 − 0.173i)2-s + (−0.642 − 0.766i)3-s + (0.939 − 0.342i)4-s + (−0.766 − 0.642i)6-s + (0.866 + 0.5i)7-s + (0.866 − 0.5i)8-s + (−0.173 + 0.984i)9-s + (−0.866 − 0.5i)12-s + (−0.642 + 0.766i)13-s + (0.939 + 0.342i)14-s + (0.766 − 0.642i)16-s + (0.984 − 0.173i)17-s + i·18-s + (−0.173 − 0.984i)21-s + (0.342 + 0.939i)23-s + (−0.939 − 0.342i)24-s + ⋯ |
L(s) = 1 | + (0.984 − 0.173i)2-s + (−0.642 − 0.766i)3-s + (0.939 − 0.342i)4-s + (−0.766 − 0.642i)6-s + (0.866 + 0.5i)7-s + (0.866 − 0.5i)8-s + (−0.173 + 0.984i)9-s + (−0.866 − 0.5i)12-s + (−0.642 + 0.766i)13-s + (0.939 + 0.342i)14-s + (0.766 − 0.642i)16-s + (0.984 − 0.173i)17-s + i·18-s + (−0.173 − 0.984i)21-s + (0.342 + 0.939i)23-s + (−0.939 − 0.342i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.887 - 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.887 - 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.588942692 - 0.6314591356i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.588942692 - 0.6314591356i\) |
\(L(1)\) |
\(\approx\) |
\(1.745382759 - 0.3953908316i\) |
\(L(1)\) |
\(\approx\) |
\(1.745382759 - 0.3953908316i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (0.984 - 0.173i)T \) |
| 3 | \( 1 + (-0.642 - 0.766i)T \) |
| 7 | \( 1 + (0.866 + 0.5i)T \) |
| 13 | \( 1 + (-0.642 + 0.766i)T \) |
| 17 | \( 1 + (0.984 - 0.173i)T \) |
| 23 | \( 1 + (0.342 + 0.939i)T \) |
| 29 | \( 1 + (0.173 - 0.984i)T \) |
| 31 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 + iT \) |
| 41 | \( 1 + (-0.766 + 0.642i)T \) |
| 43 | \( 1 + (0.342 - 0.939i)T \) |
| 47 | \( 1 + (0.984 + 0.173i)T \) |
| 53 | \( 1 + (0.342 + 0.939i)T \) |
| 59 | \( 1 + (-0.173 - 0.984i)T \) |
| 61 | \( 1 + (0.939 - 0.342i)T \) |
| 67 | \( 1 + (0.984 + 0.173i)T \) |
| 71 | \( 1 + (-0.939 - 0.342i)T \) |
| 73 | \( 1 + (0.642 + 0.766i)T \) |
| 79 | \( 1 + (0.766 - 0.642i)T \) |
| 83 | \( 1 + (-0.866 - 0.5i)T \) |
| 89 | \( 1 + (-0.766 - 0.642i)T \) |
| 97 | \( 1 + (-0.984 + 0.173i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.70135746994231357274909990628, −20.89307828317689909901034353782, −20.500550487573117356127238350651, −19.61329410318770927075637972012, −18.24758208240955463863414751939, −17.39292708827999567625409789919, −16.74574185968192101173869101619, −16.17031209784024551363664358558, −15.003858489506051788493248406314, −14.76091882370483234836278316321, −13.91778024841432556310245157991, −12.70155014986094706360147876658, −12.1972601622110158120277923011, −11.215233336119751298048004231634, −10.65522996900297589234846061384, −9.92904351854978310901935814675, −8.55884196943123605584085373804, −7.592148190907150972785718055927, −6.81000410881358791949650596815, −5.60116177944194040776998732590, −5.213109488651145270710291838151, −4.293277717936122903669071229690, −3.56950160760509250175467562566, −2.458566548431364093928594287525, −1.02120246807696004118227422802,
1.240144601943692271466244378702, 1.984201553543429717872593217393, 2.938062479706129818672685444776, 4.28064094712836361895387835099, 5.18304792580047059801268518130, 5.63903318220417823253463344457, 6.72283528403186157248760859670, 7.4175040066035652682511093529, 8.24107030300317095943842200478, 9.63649616369842346894345745924, 10.66318357287994438937694039280, 11.54317109812182219032661069556, 11.94131322481546696012366806029, 12.60185317563136409610086533584, 13.66737378819557059518625432323, 14.16540404364583822588675836723, 15.036534572009458977937992688043, 15.90252308737648190752089297372, 16.89999530758259859612478914399, 17.40887470340979726519431963181, 18.66092343681357006490476887180, 19.00447595017953887155091272145, 20.004288165435704145097625765176, 20.91752898788984845734859113991, 21.724215685492542873333187647844