| L(s) = 1 | + 0.638i·3-s − 9.06·7-s + 8.59·9-s + (10.0 − 4.36i)11-s − 10.4·13-s + 3.39·17-s + 28.0i·19-s − 5.79i·21-s − 18.8i·23-s + 11.2i·27-s − 22.2i·29-s + 0.00744·31-s + (2.79 + 6.44i)33-s + 8.96i·37-s − 6.66i·39-s + ⋯ |
| L(s) = 1 | + 0.212i·3-s − 1.29·7-s + 0.954·9-s + (0.917 − 0.397i)11-s − 0.802·13-s + 0.199·17-s + 1.47i·19-s − 0.275i·21-s − 0.818i·23-s + 0.416i·27-s − 0.766i·29-s + 0.000240·31-s + (0.0845 + 0.195i)33-s + 0.242i·37-s − 0.170i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.643 - 0.765i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.643 - 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.632821647\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.632821647\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + (-10.0 + 4.36i)T \) |
| good | 3 | \( 1 - 0.638iT - 9T^{2} \) |
| 7 | \( 1 + 9.06T + 49T^{2} \) |
| 13 | \( 1 + 10.4T + 169T^{2} \) |
| 17 | \( 1 - 3.39T + 289T^{2} \) |
| 19 | \( 1 - 28.0iT - 361T^{2} \) |
| 23 | \( 1 + 18.8iT - 529T^{2} \) |
| 29 | \( 1 + 22.2iT - 841T^{2} \) |
| 31 | \( 1 - 0.00744T + 961T^{2} \) |
| 37 | \( 1 - 8.96iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 16.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 81.5T + 1.84e3T^{2} \) |
| 47 | \( 1 - 59.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 60.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 81.5T + 3.48e3T^{2} \) |
| 61 | \( 1 - 114. iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 99.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 70.3T + 5.04e3T^{2} \) |
| 73 | \( 1 - 16.4T + 5.32e3T^{2} \) |
| 79 | \( 1 - 118. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 30.6T + 6.88e3T^{2} \) |
| 89 | \( 1 - 4.02T + 7.92e3T^{2} \) |
| 97 | \( 1 - 101. iT - 9.40e3T^{2} \) |
| 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
−9.862366709822315240464168031099, −9.193571833271259096370129444097, −8.126694423977526422470732706106, −7.15510875138820891520777119462, −6.44294363062501568149139051408, −5.66485789706827445382817851097, −4.29574911412919041434187473766, −3.70466366546393553365410159490, −2.51050121907801351062962856816, −0.992215050079552477486721689007,
0.62642778832636575409613039264, 2.07128663505076511358400092056, 3.27790679451774955311479292833, 4.21118080200690658368872352348, 5.22904073202738689602006852648, 6.45705540458713908095679216339, 6.97928697589607291009882672604, 7.61088665156285677940137587944, 9.120261585564582456800962096975, 9.415354985761376701508947541729
