L(s) = 1 | + 2-s + 4-s − i·5-s − i·7-s + 8-s − i·10-s + i·11-s + 13-s − i·14-s + 16-s − 19-s − i·20-s + i·22-s + i·23-s − 25-s + 26-s + ⋯ |
L(s) = 1 | + 2-s + 4-s − i·5-s − i·7-s + 8-s − i·10-s + i·11-s + 13-s − i·14-s + 16-s − 19-s − i·20-s + i·22-s + i·23-s − 25-s + 26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.672958108 - 0.9199004879i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.672958108 - 0.9199004879i\) |
\(L(1)\) |
\(\approx\) |
\(1.928414006 - 0.3894258837i\) |
\(L(1)\) |
\(\approx\) |
\(1.928414006 - 0.3894258837i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 17 | \( 1 \) |
good | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - iT \) |
| 19 | \( 1 - iT \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 \) |
| 31 | \( 1 - iT \) |
| 37 | \( 1 + iT \) |
| 41 | \( 1 \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 \) |
| 67 | \( 1 \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 - iT \) |
| 79 | \( 1 \) |
| 83 | \( 1 + iT \) |
| 89 | \( 1 + iT \) |
| 97 | \( 1 \) |
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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
−33.38356476329067642884693206758, −32.14927472100879125688762779413, −31.1534613843036460580488209308, −30.21113153291036467918273362675, −29.21129016915208050062693979900, −27.886087781047407570914710576367, −26.2160291616873477317017591118, −25.2610778252731973663997274707, −24.03747709746982733958881779342, −22.829811128011679122239007994697, −21.87739836701472278643370414527, −21.02439388900031901179724323399, −19.30269313911077809744978060018, −18.34024529491632444727335038199, −16.38284901242448448680710523128, −15.238200787496468529247526240717, −14.28271734391226090355093325875, −12.98976349004606714125930956926, −11.54129339458912058185009337258, −10.62523179985428436468825371641, −8.45681550538793811029420920375, −6.62713954285006385888213874461, −5.68218661108062921441775016790, −3.68846410743393866349972257376, −2.37659152792282346858494953666,
1.528753327978942981209348598759, 3.839325316752567455540691704026, 4.91554190840436925668697475861, 6.57261454046174921958129783462, 8.06637083969625612212732689440, 10.036386355670647406790283868616, 11.51163076743198454154136990214, 12.87198144277811000666930132611, 13.63831019994323071538439819604, 15.1793307920529668042419059898, 16.37201651221059475463234136141, 17.42591693534099532530297614429, 19.635677134669205131903228781850, 20.50170187630443046232750720096, 21.374684212506118429690421218209, 23.13865760431996882617606264597, 23.56189485218526766363351210505, 24.95839522664079930513399025200, 25.91665583620737115424096680749, 27.74549406695432035446513733803, 28.79218051579949608055500069202, 29.983284995421020561924176345444, 30.95960377648930974114693113450, 32.17275477310682267532508632148, 32.99791896447882294104474580319