| L(s) = 1 | + (0.741 − 1.20i)2-s + (2.28 − 0.571i)3-s + (−0.898 − 1.78i)4-s + (1.05 + 0.498i)5-s + (1.00 − 3.17i)6-s + (0.690 + 2.27i)7-s + (−2.81 − 0.243i)8-s + (2.23 − 1.19i)9-s + (1.38 − 0.898i)10-s + (−2.07 + 0.308i)11-s + (−3.07 − 3.56i)12-s + (−0.333 + 0.930i)13-s + (3.25 + 0.857i)14-s + (2.68 + 0.534i)15-s + (−2.38 + 3.21i)16-s + (−4.45 + 0.887i)17-s + ⋯ |
| L(s) = 1 | + (0.524 − 0.851i)2-s + (1.31 − 0.330i)3-s + (−0.449 − 0.893i)4-s + (0.470 + 0.222i)5-s + (0.410 − 1.29i)6-s + (0.260 + 0.860i)7-s + (−0.996 − 0.0860i)8-s + (0.745 − 0.398i)9-s + (0.436 − 0.284i)10-s + (−0.626 + 0.0930i)11-s + (−0.887 − 1.02i)12-s + (−0.0923 + 0.258i)13-s + (0.869 + 0.229i)14-s + (0.694 + 0.138i)15-s + (−0.595 + 0.802i)16-s + (−1.08 + 0.215i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.313 + 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.313 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.83734 - 1.32902i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.83734 - 1.32902i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.741 + 1.20i)T \) |
| good | 3 | \( 1 + (-2.28 + 0.571i)T + (2.64 - 1.41i)T^{2} \) |
| 5 | \( 1 + (-1.05 - 0.498i)T + (3.17 + 3.86i)T^{2} \) |
| 7 | \( 1 + (-0.690 - 2.27i)T + (-5.82 + 3.88i)T^{2} \) |
| 11 | \( 1 + (2.07 - 0.308i)T + (10.5 - 3.19i)T^{2} \) |
| 13 | \( 1 + (0.333 - 0.930i)T + (-10.0 - 8.24i)T^{2} \) |
| 17 | \( 1 + (4.45 - 0.887i)T + (15.7 - 6.50i)T^{2} \) |
| 19 | \( 1 + (2.52 + 2.28i)T + (1.86 + 18.9i)T^{2} \) |
| 23 | \( 1 + (-1.17 + 0.115i)T + (22.5 - 4.48i)T^{2} \) |
| 29 | \( 1 + (1.91 + 2.58i)T + (-8.41 + 27.7i)T^{2} \) |
| 31 | \( 1 + (-7.81 - 3.23i)T + (21.9 + 21.9i)T^{2} \) |
| 37 | \( 1 + (0.0690 - 1.40i)T + (-36.8 - 3.62i)T^{2} \) |
| 41 | \( 1 + (-5.95 + 7.26i)T + (-7.99 - 40.2i)T^{2} \) |
| 43 | \( 1 + (-0.836 + 3.34i)T + (-37.9 - 20.2i)T^{2} \) |
| 47 | \( 1 + (1.37 + 0.921i)T + (17.9 + 43.4i)T^{2} \) |
| 53 | \( 1 + (-2.16 + 2.91i)T + (-15.3 - 50.7i)T^{2} \) |
| 59 | \( 1 + (-4.19 - 11.7i)T + (-45.6 + 37.4i)T^{2} \) |
| 61 | \( 1 + (-4.79 + 7.99i)T + (-28.7 - 53.7i)T^{2} \) |
| 67 | \( 1 + (-4.80 - 2.87i)T + (31.5 + 59.0i)T^{2} \) |
| 71 | \( 1 + (6.73 - 12.5i)T + (-39.4 - 59.0i)T^{2} \) |
| 73 | \( 1 + (-2.04 + 6.74i)T + (-60.6 - 40.5i)T^{2} \) |
| 79 | \( 1 + (-9.66 - 14.4i)T + (-30.2 + 72.9i)T^{2} \) |
| 83 | \( 1 + (0.483 + 9.84i)T + (-82.6 + 8.13i)T^{2} \) |
| 89 | \( 1 + (6.13 + 0.604i)T + (87.2 + 17.3i)T^{2} \) |
| 97 | \( 1 + (7.08 - 17.1i)T + (-68.5 - 68.5i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.97600677094131839153216725069, −10.91753987840790495141561183600, −9.892466228305417836610225693492, −8.932256843286604478167618622949, −8.337669267561388190717079196316, −6.76117184463009390221927751147, −5.50784713943302858298038393093, −4.17929323904721008691467761932, −2.60529368053319427820752225905, −2.18604042994407808070154539684,
2.56582038033294007089294258178, 3.85472948380842284561769015083, 4.81220806900156312863345293499, 6.19731796647929884307972106425, 7.49806961274063103736337085783, 8.173431426227893519705415921053, 9.084704647848296424437408978038, 9.969295443200764315359251612434, 11.23969639091642336433060470031, 12.79885372034552834580541217034
