L(s) = 1 | + (−0.339 + 1.26i)2-s + i·3-s + (0.246 + 0.142i)4-s + (0.109 + 0.409i)5-s + (−1.26 − 0.339i)6-s + (−2.64 + 0.0688i)7-s + (−2.11 + 2.11i)8-s − 9-s − 0.555·10-s + (−0.991 + 0.991i)11-s + (−0.142 + 0.246i)12-s + (3.54 + 0.648i)13-s + (0.809 − 3.36i)14-s + (−0.409 + 0.109i)15-s + (−1.67 − 2.90i)16-s + (−3.47 + 6.01i)17-s + ⋯ |
L(s) = 1 | + (−0.239 + 0.894i)2-s + 0.577i·3-s + (0.123 + 0.0710i)4-s + (0.0491 + 0.183i)5-s + (−0.516 − 0.138i)6-s + (−0.999 + 0.0260i)7-s + (−0.748 + 0.748i)8-s − 0.333·9-s − 0.175·10-s + (−0.298 + 0.298i)11-s + (−0.0410 + 0.0710i)12-s + (0.983 + 0.179i)13-s + (0.216 − 0.900i)14-s + (−0.105 + 0.0283i)15-s + (−0.418 − 0.725i)16-s + (−0.842 + 1.45i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 - 0.253i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.967 - 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.127049 + 0.987690i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.127049 + 0.987690i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 7 | \( 1 + (2.64 - 0.0688i)T \) |
| 13 | \( 1 + (-3.54 - 0.648i)T \) |
good | 2 | \( 1 + (0.339 - 1.26i)T + (-1.73 - i)T^{2} \) |
| 5 | \( 1 + (-0.109 - 0.409i)T + (-4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (0.991 - 0.991i)T - 11iT^{2} \) |
| 17 | \( 1 + (3.47 - 6.01i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.391 + 0.391i)T - 19iT^{2} \) |
| 23 | \( 1 + (-6.79 + 3.92i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.01 - 5.22i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (8.09 + 2.17i)T + (26.8 + 15.5i)T^{2} \) |
| 37 | \( 1 + (-5.79 - 1.55i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-0.434 - 1.62i)T + (-35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-6.49 + 3.74i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-9.79 + 2.62i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-3.77 - 6.54i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (6.60 - 1.76i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 - 2.75iT - 61T^{2} \) |
| 67 | \( 1 + (-10.1 - 10.1i)T + 67iT^{2} \) |
| 71 | \( 1 + (-1.00 + 3.74i)T + (-61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (2.96 - 11.0i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-4.32 + 7.49i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.26 + 2.26i)T - 83iT^{2} \) |
| 89 | \( 1 + (-2.99 + 11.1i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (15.5 + 4.17i)T + (84.0 + 48.5i)T^{2} \) |
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
−12.51012693185558252922512631150, −11.03648568171146872116908244056, −10.58265002291160704295118763450, −9.080610300852050561002774038076, −8.692494254350650692148129897361, −7.27470197665246448015741626421, −6.45434761799191995607924730700, −5.61426164388197792808884434537, −4.01032151702853240486584993091, −2.71138599501762503623056061669,
0.820472095228341847157182560469, 2.53947225867514674867691211179, 3.53775202338132090183501530794, 5.50173244055385144280329192550, 6.55296325585237537006897496197, 7.42644990553408352871641135825, 9.092860632456871520750387338724, 9.404297706104456323385100468750, 10.94747740581572402891800896669, 11.17933991500793928220798493327