L(s) = 1 | − i·2-s − 4-s + i·5-s + 7-s + i·8-s + 10-s + 11-s − i·13-s − i·14-s + 16-s + i·17-s − i·19-s − i·20-s − i·22-s + i·23-s + ⋯ |
L(s) = 1 | − i·2-s − 4-s + i·5-s + 7-s + i·8-s + 10-s + 11-s − i·13-s − i·14-s + 16-s + i·17-s − i·19-s − i·20-s − i·22-s + i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9881035017 - 0.3622677591i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9881035017 - 0.3622677591i\) |
\(L(1)\) |
\(\approx\) |
\(1.005205997 - 0.3261861599i\) |
\(L(1)\) |
\(\approx\) |
\(1.005205997 - 0.3261861599i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 37 | \( 1 \) |
good | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + iT \) |
| 17 | \( 1 \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + iT \) |
| 29 | \( 1 \) |
| 31 | \( 1 + T \) |
| 41 | \( 1 \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + iT \) |
| 67 | \( 1 \) |
| 71 | \( 1 - iT \) |
| 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
−29.537090350716544791381195641623, −28.12436458621962665703466427156, −27.48179409806595153996735324015, −26.55622581463859429564152011341, −25.11055829143777148463840964884, −24.551109758737302616442485186118, −23.76250379675297104890885758263, −22.59509342469281790479679432973, −21.37418382078989191225808917594, −20.37507869049141348308132747817, −18.92717518187404463262340159150, −17.8145091671361845659028858432, −16.767852071157142961510899708689, −16.19418533316915232310880367659, −14.61908872242994616993331030442, −14.037225536500387966801553233653, −12.6173498942671991394491783829, −11.51769663487572904239078083469, −9.5744683738891522423469575518, −8.729210781987695480264678210541, −7.68783893911254707954066243435, −6.30617182653870183854456625765, −4.94835953615715578724879176005, −4.14870331785780168197330348642, −1.39957129056617761445608487221,
1.595431243656791443714747977606, 3.03592437113616164168518586034, 4.279001187352969531211547410750, 5.80436208548715236602973847442, 7.53497363543995876552634279051, 8.78371933860792514234624149618, 10.1770931808642285897416934515, 11.07911670319207838558988495661, 11.89542029520899863820477135500, 13.34612829546419814576021469075, 14.41888668486238679217372262260, 15.21026045571733586511896222171, 17.48633340862625147091893649035, 17.75109017990850829481342625104, 19.171517977172636717104438845574, 19.87747201181742139296132788829, 21.21194819607120774285783591695, 21.97315009033450599590816103648, 22.88516475180933603338049264181, 23.98528421877642107708251766347, 25.429559492161978341019745817681, 26.64510468541732578957644109433, 27.43323137913473421562073271683, 28.21248907353080012274514077345, 29.66440527821468153506502899863