L(s) = 1 | + (−0.430 + 1.60i)2-s + (1.35 − 1.08i)3-s + (−0.661 − 0.382i)4-s + (−2.23 − 0.154i)5-s + (1.15 + 2.63i)6-s + (−1.73 − 0.465i)7-s + (−1.45 + 1.45i)8-s + (0.661 − 2.92i)9-s + (1.20 − 3.51i)10-s + (3.12 − 1.80i)11-s + (−1.30 + 0.198i)12-s + (−1.27 + 0.342i)13-s + (1.49 − 2.59i)14-s + (−3.18 + 2.20i)15-s + (−2.47 − 4.28i)16-s + (0.277 + 0.277i)17-s + ⋯ |
L(s) = 1 | + (−0.304 + 1.13i)2-s + (0.781 − 0.624i)3-s + (−0.330 − 0.191i)4-s + (−0.997 − 0.0690i)5-s + (0.471 + 1.07i)6-s + (−0.656 − 0.175i)7-s + (−0.513 + 0.513i)8-s + (0.220 − 0.975i)9-s + (0.381 − 1.11i)10-s + (0.942 − 0.544i)11-s + (−0.377 + 0.0573i)12-s + (−0.354 + 0.0950i)13-s + (0.399 − 0.692i)14-s + (−0.822 + 0.568i)15-s + (−0.618 − 1.07i)16-s + (0.0671 + 0.0671i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.692211 + 0.334449i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.692211 + 0.334449i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.35 + 1.08i)T \) |
| 5 | \( 1 + (2.23 + 0.154i)T \) |
good | 2 | \( 1 + (0.430 - 1.60i)T + (-1.73 - i)T^{2} \) |
| 7 | \( 1 + (1.73 + 0.465i)T + (6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-3.12 + 1.80i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.27 - 0.342i)T + (11.2 - 6.5i)T^{2} \) |
| 17 | \( 1 + (-0.277 - 0.277i)T + 17iT^{2} \) |
| 19 | \( 1 - 6.25iT - 19T^{2} \) |
| 23 | \( 1 + (-0.579 - 2.16i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (1.56 + 2.71i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (2.42 - 4.20i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.55 + 5.55i)T - 37iT^{2} \) |
| 41 | \( 1 + (-1.29 - 0.744i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.10 - 4.10i)T + (-37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 + (1.02 - 3.82i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-7.48 + 7.48i)T - 53iT^{2} \) |
| 59 | \( 1 + (-0.279 + 0.483i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.96 + 5.13i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.90 + 10.8i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 8.01iT - 71T^{2} \) |
| 73 | \( 1 + (1.29 + 1.29i)T + 73iT^{2} \) |
| 79 | \( 1 + (-6.96 + 4.02i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (0.560 + 0.150i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + 16.4T + 89T^{2} \) |
| 97 | \( 1 + (-5.14 - 1.37i)T + (84.0 + 48.5i)T^{2} \) |
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
−16.09261190134310147601321615436, −14.92090090360207578431044602671, −14.21957815217478921101767747719, −12.65047593048595670642384387924, −11.61150115226788427953733667974, −9.398619374399456357120601865111, −8.267373598673132246039176657071, −7.35680509729583724471517328201, −6.25278543188853638197576661280, −3.55540741718480021542886524616,
2.86093392629714111854334407398, 4.21548998350390319370472016168, 7.05478013129401502391818272930, 8.839942659986665334564159960456, 9.699828025443930919117313866918, 10.92934913075462471574604889698, 11.97751315480468378099664596534, 13.11617539726927643038549088056, 14.84401014764592595659853185392, 15.48814334931586089528425035668