L(s) = 1 | + 1.41·2-s + 2.00·4-s + (−4.21 + 2.69i)5-s − 7.64i·7-s + 2.82·8-s + (−5.96 + 3.80i)10-s + 11.3i·11-s − 14.3i·13-s − 10.8i·14-s + 4.00·16-s − 17.3·17-s − 31.8·19-s + (−8.42 + 5.38i)20-s + 16.0i·22-s + 16.5·23-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.500·4-s + (−0.842 + 0.538i)5-s − 1.09i·7-s + 0.353·8-s + (−0.596 + 0.380i)10-s + 1.03i·11-s − 1.10i·13-s − 0.772i·14-s + 0.250·16-s − 1.01·17-s − 1.67·19-s + (−0.421 + 0.269i)20-s + 0.731i·22-s + 0.718·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.842 + 0.538i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.842 + 0.538i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7215337500\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7215337500\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (4.21 - 2.69i)T \) |
good | 7 | \( 1 + 7.64iT - 49T^{2} \) |
| 11 | \( 1 - 11.3iT - 121T^{2} \) |
| 13 | \( 1 + 14.3iT - 169T^{2} \) |
| 17 | \( 1 + 17.3T + 289T^{2} \) |
| 19 | \( 1 + 31.8T + 361T^{2} \) |
| 23 | \( 1 - 16.5T + 529T^{2} \) |
| 29 | \( 1 + 8.49iT - 841T^{2} \) |
| 31 | \( 1 + 45.8T + 961T^{2} \) |
| 37 | \( 1 - 31.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 68.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 31.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 46.0T + 2.20e3T^{2} \) |
| 53 | \( 1 + 53.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + 64.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 96.2T + 3.72e3T^{2} \) |
| 67 | \( 1 - 1.72iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 26.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 51.4iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 59.2T + 6.24e3T^{2} \) |
| 83 | \( 1 - 6.92T + 6.88e3T^{2} \) |
| 89 | \( 1 - 115. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 158. iT - 9.40e3T^{2} \) |
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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
−10.05822411361379811700341668768, −8.692010170582303045681474606081, −7.68108689066545170763822946722, −7.08310094692155518101080141040, −6.37424889957080892822928270496, −4.93177534963946966160392778107, −4.19375545825566777419515170358, −3.38976371159267312081144099960, −2.09119507286798294694781906565, −0.17636786838994455430398951481,
1.81176607264699408061401258005, 3.04885627097526898308432578040, 4.17448383645756226549541888208, 4.89062285749110597294741629459, 6.00155665112849996725665501544, 6.70301748706765643823160267008, 7.893920235760302876969797825200, 8.856502041716791814254720629397, 9.119634510574895655518700210568, 10.92244948403422406432156424500