L(s) = 1 | + (0.931 + 0.363i)2-s + (−0.791 − 0.611i)3-s + (0.735 + 0.678i)4-s + (0.154 + 0.987i)5-s + (−0.514 − 0.857i)6-s + (0.922 + 0.386i)7-s + (0.437 + 0.899i)8-s + (0.251 + 0.967i)9-s + (−0.215 + 0.976i)10-s + (0.545 + 0.837i)11-s + (−0.166 − 0.985i)12-s + (0.346 − 0.938i)13-s + (0.717 + 0.696i)14-s + (0.481 − 0.876i)15-s + (0.0806 + 0.996i)16-s + (−0.977 − 0.209i)17-s + ⋯ |
L(s) = 1 | + (0.931 + 0.363i)2-s + (−0.791 − 0.611i)3-s + (0.735 + 0.678i)4-s + (0.154 + 0.987i)5-s + (−0.514 − 0.857i)6-s + (0.922 + 0.386i)7-s + (0.437 + 0.899i)8-s + (0.251 + 0.967i)9-s + (−0.215 + 0.976i)10-s + (0.545 + 0.837i)11-s + (−0.166 − 0.985i)12-s + (0.346 − 0.938i)13-s + (0.717 + 0.696i)14-s + (0.481 − 0.876i)15-s + (0.0806 + 0.996i)16-s + (−0.977 − 0.209i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0956 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0956 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.556541483 + 1.414094940i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.556541483 + 1.414094940i\) |
\(L(1)\) |
\(\approx\) |
\(1.479929051 + 0.6203308014i\) |
\(L(1)\) |
\(\approx\) |
\(1.479929051 + 0.6203308014i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 \) |
good | 2 | \( 1 + (0.931 + 0.363i)T \) |
| 3 | \( 1 + (-0.791 - 0.611i)T \) |
| 5 | \( 1 + (0.154 + 0.987i)T \) |
| 7 | \( 1 + (0.922 + 0.386i)T \) |
| 11 | \( 1 + (0.545 + 0.837i)T \) |
| 13 | \( 1 + (0.346 - 0.938i)T \) |
| 17 | \( 1 + (-0.977 - 0.209i)T \) |
| 19 | \( 1 + (-0.885 + 0.465i)T \) |
| 29 | \( 1 + (0.275 - 0.961i)T \) |
| 31 | \( 1 + (-0.986 - 0.160i)T \) |
| 37 | \( 1 + (0.975 + 0.221i)T \) |
| 41 | \( 1 + (-0.381 + 0.924i)T \) |
| 43 | \( 1 + (0.867 + 0.498i)T \) |
| 47 | \( 1 + (-0.917 - 0.398i)T \) |
| 53 | \( 1 + (-0.215 - 0.976i)T \) |
| 59 | \( 1 + (0.586 - 0.809i)T \) |
| 61 | \( 1 + (0.664 + 0.747i)T \) |
| 67 | \( 1 + (0.751 - 0.659i)T \) |
| 71 | \( 1 + (0.105 + 0.994i)T \) |
| 73 | \( 1 + (0.275 + 0.961i)T \) |
| 79 | \( 1 + (-0.191 - 0.981i)T \) |
| 83 | \( 1 + (0.645 - 0.763i)T \) |
| 89 | \( 1 + (-0.926 - 0.375i)T \) |
| 97 | \( 1 + (0.948 + 0.317i)T \) |
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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
−23.54019247518826201335734977901, −22.171016079908531140873237835824, −21.61876732738286461409298165592, −21.03865529986494248839634203132, −20.26479990656383518234269259270, −19.41274773064252031372775970207, −18.09465151388152351867213807463, −17.05858210242681541838880441721, −16.47717885251169651335579205725, −15.67881300271382710938648652559, −14.64313772614526538115091526869, −13.83762127615688986562981442313, −12.89402263326956377464771867695, −12.00590268268517165259882047758, −11.12000664080565016665259928899, −10.805505606832213624999071707664, −9.36313369693574302599456740769, −8.62573546949600334662968396863, −6.93967906507238236422986012928, −6.06375883723382205287678333499, −5.14039327935677912504979427076, −4.36460721965903734395979734959, −3.821828394472526258392698651412, −1.96075137524589937816421418937, −0.94417676497474372960914473177,
1.79666798793287181866934785606, 2.50864553814832544130689605299, 4.03521508080866504008646691812, 4.98059262087156069707755950537, 5.99176989583387125169156904637, 6.6003829554704846458268283138, 7.53397145909945503196148423273, 8.2943119841056815040697951414, 10.11198008244985363964590262627, 11.23008198157966294362071710046, 11.461641151753227157545960439322, 12.646389074245462642153402218602, 13.28368094353137774612455937634, 14.434186112147963884822724231025, 14.95927375191656883500511802476, 15.82897725628864847806900506281, 17.09277334419562821240950819303, 17.721524698215896662254619665277, 18.27525016231099704140446321277, 19.50496099932325254681892094740, 20.515974063630940632640699284683, 21.57184709531391623274121235164, 22.20119191019645292661219171496, 22.918787986207758149165500352421, 23.43469893707937738989414092971