L(s) = 1 | + (−0.645 + 1.25i)2-s + (0.801 − 1.53i)3-s + (−1.16 − 1.62i)4-s + (0.5 + 0.866i)5-s + (1.41 + 1.99i)6-s + (−4.07 − 2.35i)7-s + (2.79 − 0.421i)8-s + (−1.71 − 2.46i)9-s + (−1.41 + 0.0704i)10-s + (−3.19 − 1.84i)11-s + (−3.42 + 0.489i)12-s + (−2.94 + 1.69i)13-s + (5.58 − 3.60i)14-s + (1.73 − 0.0730i)15-s + (−1.27 + 3.79i)16-s + 3.19i·17-s + ⋯ |
L(s) = 1 | + (−0.456 + 0.889i)2-s + (0.463 − 0.886i)3-s + (−0.583 − 0.812i)4-s + (0.223 + 0.387i)5-s + (0.577 + 0.816i)6-s + (−1.53 − 0.888i)7-s + (0.988 − 0.148i)8-s + (−0.571 − 0.820i)9-s + (−0.446 + 0.0222i)10-s + (−0.963 − 0.556i)11-s + (−0.989 + 0.141i)12-s + (−0.816 + 0.471i)13-s + (1.49 − 0.963i)14-s + (0.446 − 0.0188i)15-s + (−0.318 + 0.947i)16-s + 0.776i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.265424 - 0.429773i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.265424 - 0.429773i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.645 - 1.25i)T \) |
| 3 | \( 1 + (-0.801 + 1.53i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
good | 7 | \( 1 + (4.07 + 2.35i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (3.19 + 1.84i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.94 - 1.69i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 3.19iT - 17T^{2} \) |
| 19 | \( 1 + 0.184T + 19T^{2} \) |
| 23 | \( 1 + (1.68 + 2.92i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.71 + 8.16i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (1.31 - 0.761i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 2.21iT - 37T^{2} \) |
| 41 | \( 1 + (-7.97 + 4.60i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.05 + 8.75i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3.20 - 5.54i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 0.652T + 53T^{2} \) |
| 59 | \( 1 + (-7.25 + 4.19i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.59 + 3.23i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.725 + 1.25i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 11.5T + 71T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 + (2.19 + 1.26i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3.86 + 2.23i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 4.32iT - 89T^{2} \) |
| 97 | \( 1 + (8.59 - 14.8i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.76964548004477993784036235341, −10.02145740610888883623074027652, −9.222251085642095779396455726321, −8.085240950585630579710369696063, −7.30711319139641984832473766315, −6.52406489644626530101818229300, −5.84593702020888112620954578968, −3.99392951770263801561059988643, −2.52269035780856262564041701934, −0.35328194288841821356847856185,
2.54608356181009410977636827263, 3.11851502706802184366353985284, 4.63500171402487716411270931482, 5.56547656832249138250212365736, 7.33056623501287624938423557723, 8.438066045374233287556680581612, 9.366596045285562383180197934791, 9.781562996514352130348912387442, 10.45788192687712937087607699589, 11.70310252115418032984288261487