L(s) = 1 | + (0.537 + 0.843i)5-s + (0.996 − 0.0871i)7-s + (0.976 − 0.216i)11-s + (−0.737 − 0.675i)13-s + (0.866 − 0.5i)17-s + (−0.793 + 0.608i)19-s + (−0.996 − 0.0871i)23-s + (−0.422 + 0.906i)25-s + (0.0436 − 0.999i)29-s + (0.766 − 0.642i)31-s + (0.608 + 0.793i)35-s + (0.793 + 0.608i)37-s + (−0.422 − 0.906i)41-s + (−0.976 + 0.216i)43-s + (0.642 − 0.766i)47-s + ⋯ |
L(s) = 1 | + (0.537 + 0.843i)5-s + (0.996 − 0.0871i)7-s + (0.976 − 0.216i)11-s + (−0.737 − 0.675i)13-s + (0.866 − 0.5i)17-s + (−0.793 + 0.608i)19-s + (−0.996 − 0.0871i)23-s + (−0.422 + 0.906i)25-s + (0.0436 − 0.999i)29-s + (0.766 − 0.642i)31-s + (0.608 + 0.793i)35-s + (0.793 + 0.608i)37-s + (−0.422 − 0.906i)41-s + (−0.976 + 0.216i)43-s + (0.642 − 0.766i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.503 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.503 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.227189090 - 1.280475017i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.227189090 - 1.280475017i\) |
\(L(1)\) |
\(\approx\) |
\(1.322975096 - 0.05031450786i\) |
\(L(1)\) |
\(\approx\) |
\(1.322975096 - 0.05031450786i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.537 + 0.843i)T \) |
| 7 | \( 1 + (0.996 - 0.0871i)T \) |
| 11 | \( 1 + (0.976 - 0.216i)T \) |
| 13 | \( 1 + (-0.737 - 0.675i)T \) |
| 17 | \( 1 + (0.866 - 0.5i)T \) |
| 19 | \( 1 + (-0.793 + 0.608i)T \) |
| 23 | \( 1 + (-0.996 - 0.0871i)T \) |
| 29 | \( 1 + (0.0436 - 0.999i)T \) |
| 31 | \( 1 + (0.766 - 0.642i)T \) |
| 37 | \( 1 + (0.793 + 0.608i)T \) |
| 41 | \( 1 + (-0.422 - 0.906i)T \) |
| 43 | \( 1 + (-0.976 + 0.216i)T \) |
| 47 | \( 1 + (0.642 - 0.766i)T \) |
| 53 | \( 1 + (-0.382 - 0.923i)T \) |
| 59 | \( 1 + (-0.537 - 0.843i)T \) |
| 61 | \( 1 + (0.887 - 0.461i)T \) |
| 67 | \( 1 + (0.675 - 0.737i)T \) |
| 71 | \( 1 + (-0.965 - 0.258i)T \) |
| 73 | \( 1 + (0.965 - 0.258i)T \) |
| 79 | \( 1 + (-0.342 - 0.939i)T \) |
| 83 | \( 1 + (-0.999 - 0.0436i)T \) |
| 89 | \( 1 + (0.258 + 0.965i)T \) |
| 97 | \( 1 + (-0.173 + 0.984i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−20.07174843792219992517682624297, −19.76298994668502058844118594118, −18.716801842816692278393384016364, −17.83984747752117296495908979449, −17.1697147122286766959584899475, −16.808378077175483939386294198225, −15.88661959024232931644942461490, −14.75102045469578210231835641951, −14.38794455548200297575177676052, −13.630644736154440838846075713055, −12.55450564170705093131164479216, −12.092445346398874290614573974690, −11.34214427134495170020709711887, −10.29360308006164017216520533061, −9.56724803998073605438589975783, −8.776290649241035757025709984362, −8.19695776033150769477016104216, −7.18691632703716637225041553315, −6.27653382277456592062019019969, −5.42402893556217047620978983456, −4.57305311816057184481730709598, −4.088882294580344187287300446129, −2.59273834364539711549216774791, −1.65610916453047486245649373181, −1.10976932910813633407100664365,
0.45509288068066836676333193135, 1.672496449817674202821850339429, 2.37468414551630426288510383187, 3.428605932168266448591334256153, 4.30402640769557950430123604634, 5.32214615477387280051547885560, 6.07618023701069597852618331776, 6.85414722655574006002887950298, 7.844169147174707298213406615581, 8.31309562064369523597134279809, 9.67615529841597319478609977975, 10.0208362513235143183617494066, 10.947181435586421040342579469569, 11.72493246801702358288568751621, 12.28138964037178877701014949100, 13.55682878122980199216196044252, 14.07036199601515996992002648096, 14.811619731447108312613681845666, 15.16364045261492154737214597102, 16.47894628408381602054302378158, 17.263347938674298322810566905861, 17.59180252904612992312306396032, 18.61839348459557996671291558680, 19.01819709130511703295706537356, 20.083094988500139063134774942323