L(s) = 1 | − i·3-s + (0.5 − 0.866i)5-s + (−0.866 + 0.5i)7-s − 9-s + (0.866 − 0.5i)11-s + (−0.5 + 0.866i)13-s + (−0.866 − 0.5i)15-s + (0.5 + 0.866i)21-s + (0.866 + 0.5i)23-s + i·27-s + (−0.5 − 0.866i)29-s + (0.866 + 0.5i)31-s + (−0.5 − 0.866i)33-s + 0.999i·35-s + (0.866 + 0.5i)39-s + ⋯ |
L(s) = 1 | − i·3-s + (0.5 − 0.866i)5-s + (−0.866 + 0.5i)7-s − 9-s + (0.866 − 0.5i)11-s + (−0.5 + 0.866i)13-s + (−0.866 − 0.5i)15-s + (0.5 + 0.866i)21-s + (0.866 + 0.5i)23-s + i·27-s + (−0.5 − 0.866i)29-s + (0.866 + 0.5i)31-s + (−0.5 − 0.866i)33-s + 0.999i·35-s + (0.866 + 0.5i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.422 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.422 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7632674813\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7632674813\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
good | 5 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + 2iT - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
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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
−12.01921646940818089154177397630, −11.29382567476971384000956035775, −9.553639625601077482487287119231, −9.141405200033575166346468114886, −8.101475624121643854016792628220, −6.74677388518014076294327595665, −6.12897633470547152576738752297, −4.92680694316637013227842401028, −3.12918914925627571539520560095, −1.57218322916192363344311116267,
2.77031961470767503872629884662, 3.75521121491802393346092585118, 5.08644532893701287405730579469, 6.33936217767341830227216817971, 7.12433851161083276572159278385, 8.649405902211235628701397942728, 9.735895127543023512766095797925, 10.18249909288733876997531176562, 10.96788935674054466893427629186, 12.11185467201899770538948583746