L(s) = 1 | + (−0.313 + 0.543i)2-s + (−0.395 − 1.68i)3-s + (0.803 + 1.39i)4-s + (−0.166 − 0.288i)5-s + (1.03 + 0.314i)6-s + (0.5 − 0.866i)7-s − 2.26·8-s + (−2.68 + 1.33i)9-s + 0.208·10-s + (−0.5 + 0.866i)11-s + (2.02 − 1.90i)12-s + (3.15 + 5.47i)13-s + (0.313 + 0.543i)14-s + (−0.420 + 0.394i)15-s + (−0.897 + 1.55i)16-s + 2.34·17-s + ⋯ |
L(s) = 1 | + (−0.221 + 0.384i)2-s + (−0.228 − 0.973i)3-s + (0.401 + 0.695i)4-s + (−0.0744 − 0.128i)5-s + (0.424 + 0.128i)6-s + (0.188 − 0.327i)7-s − 0.799·8-s + (−0.895 + 0.444i)9-s + 0.0660·10-s + (−0.150 + 0.261i)11-s + (0.585 − 0.549i)12-s + (0.876 + 1.51i)13-s + (0.0838 + 0.145i)14-s + (−0.108 + 0.101i)15-s + (−0.224 + 0.388i)16-s + 0.568·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.690 - 0.723i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.690 - 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.19718 + 0.512680i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.19718 + 0.512680i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.395 + 1.68i)T \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
good | 2 | \( 1 + (0.313 - 0.543i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (0.166 + 0.288i)T + (-2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 + (-3.15 - 5.47i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 2.34T + 17T^{2} \) |
| 19 | \( 1 - 2.12T + 19T^{2} \) |
| 23 | \( 1 + (-1.69 - 2.93i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.13 + 1.96i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.58 - 6.20i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 0.952T + 37T^{2} \) |
| 41 | \( 1 + (1.75 + 3.03i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.05 - 3.56i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (2.17 - 3.76i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 5.73T + 53T^{2} \) |
| 59 | \( 1 + (-1.11 - 1.92i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.29 + 3.97i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.70 - 8.14i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 - 8.83T + 73T^{2} \) |
| 79 | \( 1 + (0.819 - 1.41i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (5.05 - 8.74i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 + (5.53 - 9.58i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.83900617810170355754074342928, −9.483855829046353193259105729336, −8.477661546697628108437715090978, −7.977014228939661910708543814743, −6.88104041551548730830763989024, −6.63379114259225592233000366685, −5.37700200836824373587565385087, −4.00762752137386192173366176330, −2.73737904901597907349506500635, −1.38107258486660314511348462629,
0.854520649271376930725990920575, 2.73424798719910161404405467203, 3.54621710467545481865693230777, 5.13941952747314680325428134093, 5.62484131240088257394712939916, 6.56100555156293554453153939455, 8.013714579795362371296493050083, 8.824911669892293303399536375046, 9.769665217040712301065934566716, 10.39157465819943227732476848542