| L(s) = 1 | + i·3-s − i·7-s − 9-s − 3·11-s − 4i·13-s + 6i·17-s + 4·19-s + 21-s + 3i·23-s − i·27-s − 3·29-s − 10·31-s − 3i·33-s + 7i·37-s + 4·39-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.377i·7-s − 0.333·9-s − 0.904·11-s − 1.10i·13-s + 1.45i·17-s + 0.917·19-s + 0.218·21-s + 0.625i·23-s − 0.192i·27-s − 0.557·29-s − 1.79·31-s − 0.522i·33-s + 1.15i·37-s + 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6634445224\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6634445224\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
| good | 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 3iT - 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + 10T + 31T^{2} \) |
| 37 | \( 1 - 7iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + iT - 43T^{2} \) |
| 47 | \( 1 - 12iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 - 7iT - 67T^{2} \) |
| 71 | \( 1 - 9T + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 + 17T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.557473680068734083830349075180, −8.650854741330197126309256332909, −7.81828475368748484503176337618, −7.37786403554171229777510510470, −5.99717163914688994145721380164, −5.52835759881352755659698140166, −4.60855972616640113046761949705, −3.59966988523185306072903141167, −2.93053430493917176604976833997, −1.47015112093180162887652636033,
0.22576658699438183689429287858, 1.82455179093352505784715913440, 2.65251916893691641455894072769, 3.70801032300711136872989797930, 4.98622126509834413679321234720, 5.48097599678030021515376073481, 6.54680089435997162472751484423, 7.31014788506635078761734333576, 7.78033566376799056782608340854, 8.970545750157528644032565398164
