L(s) = 1 | − 3-s + 5-s + (−1.5 + 2.59i)7-s + 9-s + (2.5 − 4.33i)11-s + (−2.5 − 4.33i)13-s − 15-s + (−1.5 − 2.59i)17-s + (0.5 + 0.866i)19-s + (1.5 − 2.59i)21-s + (−3.5 − 6.06i)23-s + 25-s − 27-s + (−2.5 + 4.33i)29-s + (−2.5 + 4.33i)31-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + (−0.566 + 0.981i)7-s + 0.333·9-s + (0.753 − 1.30i)11-s + (−0.693 − 1.20i)13-s − 0.258·15-s + (−0.363 − 0.630i)17-s + (0.114 + 0.198i)19-s + (0.327 − 0.566i)21-s + (−0.729 − 1.26i)23-s + 0.200·25-s − 0.192·27-s + (−0.464 + 0.804i)29-s + (−0.449 + 0.777i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.205i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 - 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + (8 - 1.73i)T \) |
good | 7 | \( 1 + (1.5 - 2.59i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.5 + 4.33i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.5 + 4.33i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.5 + 6.06i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.5 - 4.33i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.5 - 9.52i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.5 + 9.52i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + (6.5 - 11.2i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 14T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + (-4.5 - 7.79i)T + (-30.5 + 52.8i)T^{2} \) |
| 71 | \( 1 + (7.5 - 12.9i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.5 - 2.59i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.5 + 2.59i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (5.5 + 9.52i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + (4.5 + 7.79i)T + (-48.5 + 84.0i)T^{2} \) |
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
−8.157870091232138623343464981844, −7.20224440731163073983136388821, −6.28897632078481148633053151097, −5.92951557303000726612509437555, −5.27724550851888620062420352956, −4.37053574242863071425623343684, −3.12390699296575716746901253154, −2.65375830816605421613950706285, −1.22948114147194695122180462955, 0,
1.53753178818260995821080194644, 2.21544893410899892571228904533, 3.81185153191444415448616824335, 4.20721839715762275028841860970, 5.01352060918785277498528791384, 6.14604596856987682834053507371, 6.52091782196506608669138277283, 7.35834040630965537435261229020, 7.73784481052724786911176338015