| L(s) = 1 | + (−0.697 + 1.23i)2-s + 2.53·3-s + (−1.02 − 1.71i)4-s + i·5-s + (−1.77 + 3.12i)6-s − 0.924·7-s + (2.82 − 0.0648i)8-s + 3.43·9-s + (−1.23 − 0.697i)10-s + (3.13 + 1.07i)11-s + (−2.60 − 4.35i)12-s + 2.00·13-s + (0.644 − 1.13i)14-s + 2.53i·15-s + (−1.89 + 3.52i)16-s + 2.88i·17-s + ⋯ |
| L(s) = 1 | + (−0.493 + 0.869i)2-s + 1.46·3-s + (−0.513 − 0.858i)4-s + 0.447i·5-s + (−0.722 + 1.27i)6-s − 0.349·7-s + (0.999 − 0.0229i)8-s + 1.14·9-s + (−0.388 − 0.220i)10-s + (0.946 + 0.324i)11-s + (−0.751 − 1.25i)12-s + 0.554·13-s + (0.172 − 0.303i)14-s + 0.655i·15-s + (−0.473 + 0.880i)16-s + 0.698i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.302 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.302 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.37264 + 1.00464i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.37264 + 1.00464i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.697 - 1.23i)T \) |
| 5 | \( 1 - iT \) |
| 11 | \( 1 + (-3.13 - 1.07i)T \) |
| good | 3 | \( 1 - 2.53T + 3T^{2} \) |
| 7 | \( 1 + 0.924T + 7T^{2} \) |
| 13 | \( 1 - 2.00T + 13T^{2} \) |
| 17 | \( 1 - 2.88iT - 17T^{2} \) |
| 19 | \( 1 + 0.785iT - 19T^{2} \) |
| 23 | \( 1 - 2.25iT - 23T^{2} \) |
| 29 | \( 1 - 5.71T + 29T^{2} \) |
| 31 | \( 1 + 2.42iT - 31T^{2} \) |
| 37 | \( 1 - 3.21iT - 37T^{2} \) |
| 41 | \( 1 + 9.38iT - 41T^{2} \) |
| 43 | \( 1 + 4.76iT - 43T^{2} \) |
| 47 | \( 1 - 3.96iT - 47T^{2} \) |
| 53 | \( 1 + 2.12iT - 53T^{2} \) |
| 59 | \( 1 + 6.55T + 59T^{2} \) |
| 61 | \( 1 + 14.2T + 61T^{2} \) |
| 67 | \( 1 + 1.29T + 67T^{2} \) |
| 71 | \( 1 + 11.0iT - 71T^{2} \) |
| 73 | \( 1 - 12.7iT - 73T^{2} \) |
| 79 | \( 1 - 10.6T + 79T^{2} \) |
| 83 | \( 1 + 14.9iT - 83T^{2} \) |
| 89 | \( 1 - 3.77T + 89T^{2} \) |
| 97 | \( 1 + 5.21T + 97T^{2} \) |
| 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
−10.95887039461291011260466654039, −9.988009749852838475093780313231, −9.249747764216247901534873937360, −8.592905154814277210086970448899, −7.75643318245033415698774770227, −6.86113436720744604787739285943, −6.00935570827401257452339080257, −4.34571724834322790396408530743, −3.32410121183042167662835532242, −1.74689927507107844559057297258,
1.33690249056500642621600091948, 2.75819529077586887389386926632, 3.58832955560192383880809096348, 4.61660420005744690685845766901, 6.50647289513760229789635105688, 7.78132824830022115333721774855, 8.483906362528592842744796847865, 9.187599468237406644677838174867, 9.700034163316044259928478113172, 10.83953771852600288405148516403
