L(s) = 1 | − 1.41i·2-s − 2.00·4-s + 4i·5-s + 2.82i·8-s + 5.65·10-s + 2.82i·11-s − 12.7·13-s + 4.00·16-s + 4i·17-s − 22.6·19-s − 8.00i·20-s + 4.00·22-s − 36.7i·23-s + 9·25-s + 18i·26-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.500·4-s + 0.800i·5-s + 0.353i·8-s + 0.565·10-s + 0.257i·11-s − 0.979·13-s + 0.250·16-s + 0.235i·17-s − 1.19·19-s − 0.400i·20-s + 0.181·22-s − 1.59i·23-s + 0.359·25-s + 0.692i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7979295779\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7979295779\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 4iT - 25T^{2} \) |
| 11 | \( 1 - 2.82iT - 121T^{2} \) |
| 13 | \( 1 + 12.7T + 169T^{2} \) |
| 17 | \( 1 - 4iT - 289T^{2} \) |
| 19 | \( 1 + 22.6T + 361T^{2} \) |
| 23 | \( 1 + 36.7iT - 529T^{2} \) |
| 29 | \( 1 + 32.5iT - 841T^{2} \) |
| 31 | \( 1 - 50.9T + 961T^{2} \) |
| 37 | \( 1 + 32T + 1.36e3T^{2} \) |
| 41 | \( 1 + 38iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 20T + 1.84e3T^{2} \) |
| 47 | \( 1 + 20iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 94.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 83.4T + 3.72e3T^{2} \) |
| 67 | \( 1 + 48T + 4.48e3T^{2} \) |
| 71 | \( 1 + 76.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 120.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 148T + 6.24e3T^{2} \) |
| 83 | \( 1 - 80iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 106iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 154.T + 9.40e3T^{2} \) |
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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
−9.961454153195400339750903717766, −8.819066983848661291342418107018, −8.059754946288008646145346550742, −6.94270571434340631402145303890, −6.25044464049214110665851980855, −4.89358714918937044949945249918, −4.11187377432161820979494012702, −2.82402918858490982687021733281, −2.12389168712459658678415954729, −0.27229488788256116857157637041,
1.28140172616262453753795129192, 2.93147448527198877690404276892, 4.33275798585194788503316457314, 5.00481483976617660077083118511, 5.90900750799775535169453624493, 6.90820627297090528649355359121, 7.73838104132399426682349088712, 8.586597518087836546987212597750, 9.237522084635056143512448117155, 10.05948431790276342903689149872