L(s) = 1 | − 2-s + 4-s + 5-s + 7-s − 8-s − 10-s + 11-s + 13-s − 14-s + 16-s + 17-s − 19-s + 20-s − 22-s − 23-s + 25-s − 26-s + 28-s − 29-s − 31-s − 32-s − 34-s + 35-s + 37-s + 38-s − 40-s + 41-s + ⋯ |
L(s) = 1 | − 2-s + 4-s + 5-s + 7-s − 8-s − 10-s + 11-s + 13-s − 14-s + 16-s + 17-s − 19-s + 20-s − 22-s − 23-s + 25-s − 26-s + 28-s − 29-s − 31-s − 32-s − 34-s + 35-s + 37-s + 38-s − 40-s + 41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2019 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2019 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.581759190\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.581759190\) |
\(L(1)\) |
\(\approx\) |
\(1.118669238\) |
\(L(1)\) |
\(\approx\) |
\(1.118669238\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 673 | \( 1 \) |
good | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + T \) |
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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
−19.8015645915458631417236615256, −18.66812487480450238251198694440, −18.34175966316775643777543217581, −17.61131128761256305472681788746, −16.86969657168099027968402922196, −16.56636221992063874010865368155, −15.41616750993446863324646579184, −14.522791865907273324694216115757, −14.23219378342894819785299418909, −13.06149102939022830462856885835, −12.22561214969561943352547835306, −11.31685938720214589137189596794, −10.843756846461579454460904386541, −9.98332896456712124495774408030, −9.25883937032673340690216918466, −8.6196149863422329718514240429, −7.88845807713250847623716069622, −7.004695145879791837463799600382, −5.991303701433148570803290356358, −5.73581308263650638240690533043, −4.333497275689258812742246731256, −3.36152232368201662050587027745, −2.07498195617091064987124074658, −1.636642357316434004182181599147, −0.80118824758378985412761499937,
0.80118824758378985412761499937, 1.636642357316434004182181599147, 2.07498195617091064987124074658, 3.36152232368201662050587027745, 4.333497275689258812742246731256, 5.73581308263650638240690533043, 5.991303701433148570803290356358, 7.004695145879791837463799600382, 7.88845807713250847623716069622, 8.6196149863422329718514240429, 9.25883937032673340690216918466, 9.98332896456712124495774408030, 10.843756846461579454460904386541, 11.31685938720214589137189596794, 12.22561214969561943352547835306, 13.06149102939022830462856885835, 14.23219378342894819785299418909, 14.522791865907273324694216115757, 15.41616750993446863324646579184, 16.56636221992063874010865368155, 16.86969657168099027968402922196, 17.61131128761256305472681788746, 18.34175966316775643777543217581, 18.66812487480450238251198694440, 19.8015645915458631417236615256