L(s) = 1 | − 2-s + (1.72 − 0.0900i)3-s + 4-s + (−1.98 + 1.14i)5-s + (−1.72 + 0.0900i)6-s + (−0.877 + 2.49i)7-s − 8-s + (2.98 − 0.311i)9-s + (1.98 − 1.14i)10-s + (−0.148 − 0.257i)11-s + (1.72 − 0.0900i)12-s + (3.20 − 1.66i)13-s + (0.877 − 2.49i)14-s + (−3.32 + 2.15i)15-s + 16-s + 0.893·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.998 − 0.0519i)3-s + 0.5·4-s + (−0.886 + 0.511i)5-s + (−0.706 + 0.0367i)6-s + (−0.331 + 0.943i)7-s − 0.353·8-s + (0.994 − 0.103i)9-s + (0.626 − 0.361i)10-s + (−0.0447 − 0.0775i)11-s + (0.499 − 0.0259i)12-s + (0.887 − 0.460i)13-s + (0.234 − 0.667i)14-s + (−0.858 + 0.557i)15-s + 0.250·16-s + 0.216·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.190 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.190 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.887425 + 0.731897i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.887425 + 0.731897i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (-1.72 + 0.0900i)T \) |
| 7 | \( 1 + (0.877 - 2.49i)T \) |
| 13 | \( 1 + (-3.20 + 1.66i)T \) |
good | 5 | \( 1 + (1.98 - 1.14i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.148 + 0.257i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 - 0.893T + 17T^{2} \) |
| 19 | \( 1 + (3.94 - 6.83i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 7.81iT - 23T^{2} \) |
| 29 | \( 1 + (0.980 + 0.566i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.839 - 1.45i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 4.99iT - 37T^{2} \) |
| 41 | \( 1 + (-6.52 - 3.76i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.94 + 3.36i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.21 + 3.00i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (6.28 + 3.62i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 6.02iT - 59T^{2} \) |
| 61 | \( 1 + (-7.31 - 4.22i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.94 + 1.69i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (1.14 + 1.99i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-6.16 + 10.6i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.46 - 7.73i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 1.54iT - 83T^{2} \) |
| 89 | \( 1 + 14.3iT - 89T^{2} \) |
| 97 | \( 1 + (1.19 + 2.06i)T + (-48.5 + 84.0i)T^{2} \) |
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
−10.86118532261291517142750223644, −9.936917557767584700625370316037, −9.144846135606178789230822953377, −8.176568057026644260194670963078, −7.86436417011213450633194873251, −6.71038867387877187702723882394, −5.67986141667151571126137865378, −3.78846177162922805676020669514, −3.16936262659960466474962763089, −1.77526722413401910996639005726,
0.77207983010443259117171637343, 2.49296726288061717287599045765, 3.87310626159576129282521015627, 4.48770109498329317875578592369, 6.48812568387395564564688144999, 7.24874960257806598260597403658, 8.119566717810452932349341024310, 8.772981533757668136879706113339, 9.477928927841597143678745979895, 10.61723294110689084502011608479