L(s) = 1 | + 1.73i·3-s + (1 + 1.73i)4-s + (2.5 + 0.866i)7-s − 2.99·9-s + (−2.99 + 1.73i)12-s + (−1 − 3.46i)13-s + (−1.99 + 3.46i)16-s + 19-s + (−1.49 + 4.33i)21-s + (−2.5 + 4.33i)25-s − 5.19i·27-s + (1.00 + 5.19i)28-s + (3.5 − 6.06i)31-s + (−2.99 − 5.19i)36-s + (4.5 + 2.59i)37-s + ⋯ |
L(s) = 1 | + 0.999i·3-s + (0.5 + 0.866i)4-s + (0.944 + 0.327i)7-s − 0.999·9-s + (−0.866 + 0.499i)12-s + (−0.277 − 0.960i)13-s + (−0.499 + 0.866i)16-s + 0.229·19-s + (−0.327 + 0.944i)21-s + (−0.5 + 0.866i)25-s − 0.999i·27-s + (0.188 + 0.981i)28-s + (0.628 − 1.08i)31-s + (−0.499 − 0.866i)36-s + (0.739 + 0.427i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0614 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0614 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.990260 + 1.05310i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.990260 + 1.05310i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 1.73iT \) |
| 7 | \( 1 + (-2.5 - 0.866i)T \) |
| 13 | \( 1 + (1 + 3.46i)T \) |
good | 2 | \( 1 + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.5 + 6.06i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.5 - 2.59i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4 + 6.92i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + 15.5iT - 61T^{2} \) |
| 67 | \( 1 - 12.1iT - 67T^{2} \) |
| 71 | \( 1 + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-5 + 8.66i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (6.5 + 11.2i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (9.5 - 16.4i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.84279102773458749891848718418, −11.29987303327754058531531549449, −10.38807944101643464127052967140, −9.257672160291920509980622102825, −8.235021709138087269948939634017, −7.58221246124543804927297475977, −5.96524646444371478154392917413, −4.89222525149688574510108660161, −3.70173792585429273771924206217, −2.49036549984100470823466325572,
1.26970395328331028953188725527, 2.43711247051589319164585414871, 4.57562646983351975733300110648, 5.77282430805161703084340158019, 6.75189438300436116532998695716, 7.56917064314693877112065198082, 8.636590635276103521696480366800, 9.845918452361718278981875229179, 10.98105627396780032222689239593, 11.59274146321204090495889371705