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 + 8.66i·19-s + (−1.49 − 4.33i)21-s + (2.5 + 4.33i)25-s − 5.19i·27-s + (1.00 − 5.19i)28-s + (−7.5 + 4.33i)31-s + (2.99 − 5.19i)36-s + (5.5 + 9.52i)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 + 1.98i·19-s + (−0.327 − 0.944i)21-s + (0.5 + 0.866i)25-s − 0.999i·27-s + (0.188 − 0.981i)28-s + (−1.34 + 0.777i)31-s + (0.499 − 0.866i)36-s + (0.904 + 1.56i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.941 - 0.335i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.941 - 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.129743 + 0.750152i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.129743 + 0.750152i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 8.66iT - 19T^{2} \) |
| 23 | \( 1 + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (7.5 - 4.33i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.5 - 9.52i)T + (-18.5 + 32.0i)T^{2} \) |
| 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 | \( 1 - 11T + 67T^{2} \) |
| 71 | \( 1 + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (12 - 6.92i)T + (36.5 - 63.2i)T^{2} \) |
| 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 | \( 1 + (-4.5 + 2.59i)T + (48.5 - 84.0i)T^{2} \) |
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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
−12.47108013941927306407568050761, −11.35762938656492406484639141040, −10.23319924006779617837292489113, −9.514507975512239538876784958308, −8.604247727463603858204972038708, −7.74223007997395040480041342616, −6.15648242665337708955878744780, −5.11600618082989964288531664496, −3.71976796117377401600108823055, −3.10498797724630801994885533464,
0.58208706017827288621124243378, 2.38166414081353019223230660418, 4.16755189048506287075935004495, 5.60374273128433385286993151283, 6.55919735561680489925539031100, 7.27174023488180175545848110315, 8.886356997181741808525570788830, 9.311893064436845493695050786215, 10.65370307297799945834805003725, 11.42376862483112469163445446794