L(s) = 1 | + (1.23 + 2.13i)5-s + (0.5 − 0.866i)7-s + (2.32 − 4.02i)11-s + (−3.55 − 6.15i)13-s + 4.51·17-s + 4.32·19-s + (2.93 + 5.08i)23-s + (−0.527 + 0.912i)25-s + (−3.48 + 6.04i)29-s + (3.69 + 6.39i)31-s + 2.46·35-s − 0.726·37-s + (0.136 + 0.236i)41-s + (2.41 − 4.18i)43-s + (1.83 − 3.18i)47-s + ⋯ |
L(s) = 1 | + (0.550 + 0.952i)5-s + (0.188 − 0.327i)7-s + (0.700 − 1.21i)11-s + (−0.985 − 1.70i)13-s + 1.09·17-s + 0.992·19-s + (0.611 + 1.05i)23-s + (−0.105 + 0.182i)25-s + (−0.647 + 1.12i)29-s + (0.662 + 1.14i)31-s + 0.415·35-s − 0.119·37-s + (0.0213 + 0.0369i)41-s + (0.368 − 0.638i)43-s + (0.267 − 0.463i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.128i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.128i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.78636 - 0.115343i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.78636 - 0.115343i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
good | 5 | \( 1 + (-1.23 - 2.13i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.32 + 4.02i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (3.55 + 6.15i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 4.51T + 17T^{2} \) |
| 19 | \( 1 - 4.32T + 19T^{2} \) |
| 23 | \( 1 + (-2.93 - 5.08i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.48 - 6.04i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.69 - 6.39i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 0.726T + 37T^{2} \) |
| 41 | \( 1 + (-0.136 - 0.236i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.41 + 4.18i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.83 + 3.18i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 5.05T + 53T^{2} \) |
| 59 | \( 1 + (-4.56 - 7.90i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.90 + 11.9i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.663 - 1.14i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 + 4.32T + 73T^{2} \) |
| 79 | \( 1 + (3.21 - 5.57i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.742 + 1.28i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 9.83T + 89T^{2} \) |
| 97 | \( 1 + (-0.246 + 0.426i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.31895669784349060948171191769, −9.706001783510572742139133314222, −8.596944568385859470432232738890, −7.58042181518562402398983495248, −6.96753043642573184459411007801, −5.73987129356478244018004058535, −5.24795812566315373494790228231, −3.40300661289872818950601619119, −2.98331655454205316433696737050, −1.12859671494676234831687650753,
1.38284535622785182245152206167, 2.42723228903060224390699962440, 4.24079307093870095856474568340, 4.82595423664910743145180421057, 5.84520855778755482136443900236, 6.92689631573236672541201299348, 7.72587829993190728758704632380, 8.928098947220523875391857130569, 9.552026435431906320660366567632, 9.914815566210789384195134755279