L(s) = 1 | + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + (0.866 − 0.5i)5-s + i·8-s + 10-s + i·11-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s − i·19-s + (0.866 + 0.5i)20-s + (−0.5 + 0.866i)22-s + (−0.5 + 0.866i)23-s + (0.5 − 0.866i)25-s + (0.5 + 0.866i)29-s + (0.866 + 0.5i)31-s + (−0.866 + 0.5i)32-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + (0.866 − 0.5i)5-s + i·8-s + 10-s + i·11-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s − i·19-s + (0.866 + 0.5i)20-s + (−0.5 + 0.866i)22-s + (−0.5 + 0.866i)23-s + (0.5 − 0.866i)25-s + (0.5 + 0.866i)29-s + (0.866 + 0.5i)31-s + (−0.866 + 0.5i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.868702396 + 2.592771865i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.868702396 + 2.592771865i\) |
\(L(1)\) |
\(\approx\) |
\(1.895033399 + 0.8522740537i\) |
\(L(1)\) |
\(\approx\) |
\(1.895033399 + 0.8522740537i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 + (0.866 - 0.5i)T \) |
| 11 | \( 1 + iT \) |
| 17 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 - iT \) |
| 23 | \( 1 + (-0.5 + 0.866i)T \) |
| 29 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 + (0.866 + 0.5i)T \) |
| 37 | \( 1 + (0.866 + 0.5i)T \) |
| 41 | \( 1 + (0.866 - 0.5i)T \) |
| 43 | \( 1 + (0.5 - 0.866i)T \) |
| 47 | \( 1 + (-0.866 + 0.5i)T \) |
| 53 | \( 1 + (0.5 - 0.866i)T \) |
| 59 | \( 1 + (-0.866 + 0.5i)T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 + iT \) |
| 71 | \( 1 + (0.866 + 0.5i)T \) |
| 73 | \( 1 + (-0.866 - 0.5i)T \) |
| 79 | \( 1 + (-0.5 - 0.866i)T \) |
| 83 | \( 1 + iT \) |
| 89 | \( 1 + (0.866 + 0.5i)T \) |
| 97 | \( 1 + (0.866 + 0.5i)T \) |
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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
−24.84183414526034125516176557854, −24.5639092038204155580284867742, −23.07791844573888249829879190582, −22.624123147478025709141083104, −21.43840960860212160833469299408, −21.12654605945714852106170364060, −19.96818141803840240836684267656, −18.82820929439378373126466429784, −18.28289554480045929856840367810, −16.80952129307369137465061777706, −15.91455110760345606729513190693, −14.60952416146965299109982135577, −14.04332906147355416869053329591, −13.25592969505264651417343221769, −12.122707149222101620254188754129, −11.152550092097868727571328061948, −10.23477391292583670437245656938, −9.42030175896734844762027512781, −7.8117955638848116619902167107, −6.31172738739639763160573980088, −5.85959286198268077424986805380, −4.54621337916644065875350981656, −3.215648524234164141976847882325, −2.34932483927114060945175230178, −0.92332999332860958946714688452,
1.57533493006363526631362010773, 2.782591820315028072588106537220, 4.25906332805877549553704869622, 5.15159806153537778926053131862, 6.12239646825979226048761690154, 7.13399821423469562081335317511, 8.29901847814622723659650144934, 9.43372965907410280497856821654, 10.57948952900994810281831423004, 11.97348055222287366255885706760, 12.74184188162082192408353009416, 13.55716496548867467646395287306, 14.471176482790033792629714606629, 15.407232915102413955519278824993, 16.36676903227321874392496985705, 17.4167175360859847163449950383, 17.828633129109673652402081174601, 19.6056919468134789155503780469, 20.47568137976394670053365042906, 21.39677196855784119602169186307, 21.96933622900060921689543529521, 23.08294214486952908624980706718, 23.90363051956152312342991778165, 24.70744530314254500486948810464, 25.80446708932654647693909341494