L(s) = 1 | + (−0.370 − 0.928i)2-s + (−0.994 + 0.108i)3-s + (−0.725 + 0.687i)4-s + (0.647 + 0.762i)5-s + (0.468 + 0.883i)6-s + (−0.856 + 0.515i)7-s + (0.907 + 0.419i)8-s + (0.976 − 0.214i)9-s + (0.468 − 0.883i)10-s + (0.0541 + 0.998i)11-s + (0.647 − 0.762i)12-s + (0.976 + 0.214i)13-s + (0.796 + 0.605i)14-s + (−0.725 − 0.687i)15-s + (0.0541 − 0.998i)16-s + (−0.856 − 0.515i)17-s + ⋯ |
L(s) = 1 | + (−0.370 − 0.928i)2-s + (−0.994 + 0.108i)3-s + (−0.725 + 0.687i)4-s + (0.647 + 0.762i)5-s + (0.468 + 0.883i)6-s + (−0.856 + 0.515i)7-s + (0.907 + 0.419i)8-s + (0.976 − 0.214i)9-s + (0.468 − 0.883i)10-s + (0.0541 + 0.998i)11-s + (0.647 − 0.762i)12-s + (0.976 + 0.214i)13-s + (0.796 + 0.605i)14-s + (−0.725 − 0.687i)15-s + (0.0541 − 0.998i)16-s + (−0.856 − 0.515i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.884 + 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.884 + 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4978743354 + 0.1232128460i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4978743354 + 0.1232128460i\) |
\(L(1)\) |
\(\approx\) |
\(0.6292154807 + 0.01358498255i\) |
\(L(1)\) |
\(\approx\) |
\(0.6292154807 + 0.01358498255i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 59 | \( 1 \) |
good | 2 | \( 1 + (-0.370 - 0.928i)T \) |
| 3 | \( 1 + (-0.994 + 0.108i)T \) |
| 5 | \( 1 + (0.647 + 0.762i)T \) |
| 7 | \( 1 + (-0.856 + 0.515i)T \) |
| 11 | \( 1 + (0.0541 + 0.998i)T \) |
| 13 | \( 1 + (0.976 + 0.214i)T \) |
| 17 | \( 1 + (-0.856 - 0.515i)T \) |
| 19 | \( 1 + (0.267 + 0.963i)T \) |
| 23 | \( 1 + (-0.561 + 0.827i)T \) |
| 29 | \( 1 + (-0.370 + 0.928i)T \) |
| 31 | \( 1 + (0.267 - 0.963i)T \) |
| 37 | \( 1 + (0.907 - 0.419i)T \) |
| 41 | \( 1 + (-0.561 - 0.827i)T \) |
| 43 | \( 1 + (0.0541 - 0.998i)T \) |
| 47 | \( 1 + (0.647 - 0.762i)T \) |
| 53 | \( 1 + (0.468 + 0.883i)T \) |
| 61 | \( 1 + (-0.370 - 0.928i)T \) |
| 67 | \( 1 + (0.907 + 0.419i)T \) |
| 71 | \( 1 + (0.647 - 0.762i)T \) |
| 73 | \( 1 + (0.796 + 0.605i)T \) |
| 79 | \( 1 + (-0.994 - 0.108i)T \) |
| 83 | \( 1 + (-0.947 - 0.319i)T \) |
| 89 | \( 1 + (-0.370 + 0.928i)T \) |
| 97 | \( 1 + (0.796 - 0.605i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−32.685063793923200054186868656345, −32.30974257468215292355411162934, −30.20955061461945335439454917822, −28.743929684169540768513809832152, −28.450988282597575219733368702173, −26.96943696680670727384372303306, −25.8993757242072715985572398496, −24.57850493694558722651266917420, −23.84529186374458858309019503025, −22.69717581878790941376380220789, −21.637330335666023234309120824957, −19.81012208632718608221243109909, −18.422799992097695922772714067570, −17.384648763754007799330367824874, −16.46512713693126997560170746274, −15.78121969008632255387516201855, −13.619781508236490369493553613686, −12.99935293261644459188029780701, −10.950632726270927312893068613049, −9.74821603493718014808713138140, −8.42207853705738286628070569021, −6.56412367876226944279211773137, −5.89578524825557973753828141448, −4.40382969252476865015841139476, −0.88983617026657682764085744376,
1.993552558425540527868224092198, 3.80255226338710792037644315788, 5.67863225899467707527916255108, 7.06549992544988261202405965075, 9.32786274618218436440787053855, 10.17400058094007107495507536911, 11.32655719257299049834575505785, 12.4739193561578331121823484895, 13.623336947779849447792802694938, 15.57913909620585078953592893992, 16.96738995190684659204411387816, 18.16910748307692375024129238531, 18.642713051365525527076253402461, 20.34161718107479655204892672639, 21.65667047777952049855189304752, 22.41510722130485289957082748264, 23.1715538202585891142319989468, 25.314632331084026524692971904377, 26.253878630956149120426165409227, 27.57422583375618220519881071043, 28.59307483409685249024462963701, 29.24168353068230502842569995296, 30.2446186083916687120720766899, 31.41886951577264295270699024811, 33.01519086716472907180596900143