L(s) = 1 | + 2-s + (0.707 − 0.707i)3-s + 4-s + (0.707 − 0.707i)6-s + (−0.707 − 0.707i)7-s + 8-s − i·9-s + (0.707 − 0.707i)11-s + (0.707 − 0.707i)12-s + i·13-s + (−0.707 − 0.707i)14-s + 16-s − i·18-s + i·19-s − 21-s + (0.707 − 0.707i)22-s + ⋯ |
L(s) = 1 | + 2-s + (0.707 − 0.707i)3-s + 4-s + (0.707 − 0.707i)6-s + (−0.707 − 0.707i)7-s + 8-s − i·9-s + (0.707 − 0.707i)11-s + (0.707 − 0.707i)12-s + i·13-s + (−0.707 − 0.707i)14-s + 16-s − i·18-s + i·19-s − 21-s + (0.707 − 0.707i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.564 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.564 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.306900455 - 1.744140732i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.306900455 - 1.744140732i\) |
\(L(1)\) |
\(\approx\) |
\(2.234973436 - 0.7061374073i\) |
\(L(1)\) |
\(\approx\) |
\(2.234973436 - 0.7061374073i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 17 | \( 1 \) |
good | 2 | \( 1 + T \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
| 11 | \( 1 + (0.707 - 0.707i)T \) |
| 13 | \( 1 + iT \) |
| 19 | \( 1 + iT \) |
| 23 | \( 1 + (-0.707 - 0.707i)T \) |
| 29 | \( 1 + (0.707 + 0.707i)T \) |
| 31 | \( 1 + (0.707 + 0.707i)T \) |
| 37 | \( 1 + (-0.707 + 0.707i)T \) |
| 41 | \( 1 + (-0.707 + 0.707i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 - iT \) |
| 61 | \( 1 + (-0.707 + 0.707i)T \) |
| 67 | \( 1 + iT \) |
| 71 | \( 1 + (0.707 + 0.707i)T \) |
| 73 | \( 1 + (-0.707 + 0.707i)T \) |
| 79 | \( 1 + (-0.707 + 0.707i)T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 + (0.707 - 0.707i)T \) |
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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
−30.803420747922941451576663099550, −30.00601408314237265171870599325, −28.527060122286135654851993800105, −27.64446895151106455123147691495, −25.98781509983757350214017751355, −25.363351680766149564240474387558, −24.43181994537158636163895765089, −22.74065357076088524610104776546, −22.19377360722073305303821461771, −21.13085135317110113969094418398, −20.01922902400411570913054640596, −19.32801183786490852736557904533, −17.316175513184322518768937038895, −15.713283978378061598142647827279, −15.40134370978321734058756182921, −14.13992717643873104463026589239, −13.03871980294941738361869021479, −11.885869098630981034143982957880, −10.38847760440593726111994776272, −9.24840086776312417597462176282, −7.65719490785394029755611514586, −6.09872388087155823967100055449, −4.748148868584009453015508750209, −3.457956661405640418945804152221, −2.32968474890305295936909797420,
1.43644379061914263667458918564, 3.10861623608053527279576182566, 4.13876931827194811027384660645, 6.249327812753090324676014694353, 6.9590735885187606061305381916, 8.4471260596983335143522506459, 10.10143157917135555912863423860, 11.72991286987807773182716123822, 12.68909151167478362807786928784, 13.94675993836388078814453022064, 14.27916791656150432749130452834, 15.9420948523297909841040665654, 16.94475797560516927502120951337, 18.835342328822274740964565326255, 19.662288220205569546105109941727, 20.59223568789344073716795839729, 21.79315992125735158432127600198, 23.02876699311268419994810879075, 23.91964861408058195547481836586, 24.83026959507889161801752512002, 25.826885276448170752436845670471, 26.80198354864851235451183767947, 28.82899249986115262470030821011, 29.56352946201280617642286049480, 30.37949726527011483859753243971