L(s) = 1 | + 3.31i·2-s + 3·3-s − 7·4-s − 6.63i·5-s + 9.94i·6-s − 8·7-s − 9.94i·8-s + 9·9-s + 22·10-s + 3.31i·11-s − 21·12-s + 4·13-s − 26.5i·14-s − 19.8i·15-s + 5.00·16-s + 13.2i·17-s + ⋯ |
L(s) = 1 | + 1.65i·2-s + 3-s − 1.75·4-s − 1.32i·5-s + 1.65i·6-s − 1.14·7-s − 1.24i·8-s + 9-s + 2.20·10-s + 0.301i·11-s − 1.75·12-s + 0.307·13-s − 1.89i·14-s − 1.32i·15-s + 0.312·16-s + 0.780i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.813379 + 0.813379i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.813379 + 0.813379i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 3T \) |
| 11 | \( 1 - 3.31iT \) |
good | 2 | \( 1 - 3.31iT - 4T^{2} \) |
| 5 | \( 1 + 6.63iT - 25T^{2} \) |
| 7 | \( 1 + 8T + 49T^{2} \) |
| 13 | \( 1 - 4T + 169T^{2} \) |
| 17 | \( 1 - 13.2iT - 289T^{2} \) |
| 19 | \( 1 + 6T + 361T^{2} \) |
| 23 | \( 1 - 6.63iT - 529T^{2} \) |
| 29 | \( 1 + 39.7iT - 841T^{2} \) |
| 31 | \( 1 + 26T + 961T^{2} \) |
| 37 | \( 1 - 30T + 1.36e3T^{2} \) |
| 41 | \( 1 - 13.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 42T + 1.84e3T^{2} \) |
| 47 | \( 1 - 86.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 59.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 66.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 12T + 3.72e3T^{2} \) |
| 67 | \( 1 - 2T + 4.48e3T^{2} \) |
| 71 | \( 1 + 59.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 74T + 5.32e3T^{2} \) |
| 79 | \( 1 + 40T + 6.24e3T^{2} \) |
| 83 | \( 1 + 39.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 119. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 62T + 9.40e3T^{2} \) |
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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
−16.39340489348794926799824590827, −15.78213262934442172356563398244, −14.71618178752052947563904800675, −13.34101850161211493925304591171, −12.77656968448705242970921055972, −9.625344277890424690454082788157, −8.782121035391266122095343451094, −7.68385747545354478235407501398, −6.11198907827616590669215168131, −4.29685542700832488628962029342,
2.68129674928404865106074845948, 3.62327587915032566364728411320, 6.93970335167476522991480177841, 9.026354364107742844429786346354, 10.07999574161445947999091575522, 11.00398082884895023266604412421, 12.57513390229127967014706317425, 13.56469871033383997019074328678, 14.56671741647811408569175428269, 16.01398418524725385160950626319