L(s) = 1 | + (0.707 + 1.22i)2-s + (−0.999 + 1.73i)4-s + (7.24 − 4.18i)5-s − 2.82·8-s + (10.2 + 5.91i)10-s + (3 − 5.19i)11-s − 17.8i·13-s + (−2.00 − 3.46i)16-s + (−16.2 − 9.37i)17-s + (14.7 − 8.51i)19-s + 16.7i·20-s + 8.48·22-s + (6.72 + 11.6i)23-s + (22.4 − 38.9i)25-s + (21.8 − 12.6i)26-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (1.44 − 0.836i)5-s − 0.353·8-s + (1.02 + 0.591i)10-s + (0.272 − 0.472i)11-s − 1.37i·13-s + (−0.125 − 0.216i)16-s + (−0.955 − 0.551i)17-s + (0.775 − 0.447i)19-s + 0.836i·20-s + 0.385·22-s + (0.292 + 0.506i)23-s + (0.898 − 1.55i)25-s + (0.841 − 0.485i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.669850337\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.669850337\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 1.22i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-7.24 + 4.18i)T + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (-3 + 5.19i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + 17.8iT - 169T^{2} \) |
| 17 | \( 1 + (16.2 + 9.37i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-14.7 + 8.51i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-6.72 - 11.6i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + 33.9T + 841T^{2} \) |
| 31 | \( 1 + (12.7 + 7.37i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (2.98 + 5.17i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + 35.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 15.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + (28.7 - 16.6i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (17.2 - 29.9i)T + (-1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-23.6 - 13.6i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-34.9 + 20.1i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-57.1 + 99.0i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 18.6T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-101. - 58.5i)T + (2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-44.1 - 76.5i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 75.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (18 - 10.3i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 - 30.5iT - 9.40e3T^{2} \) |
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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
−9.464856363610213038090757908475, −9.186821389813148355065762532079, −8.188506803910121360122490975134, −7.21116730850747500706771789337, −6.16522410797379262053808125077, −5.45220163431749512709517449049, −4.94296737508998229011540241242, −3.50313052271167810548663863676, −2.25332645593227090224806523461, −0.76606488849368643904385243848,
1.66822693466611157427724257612, 2.24961268277246578405710696529, 3.50979265591099789635513051321, 4.61541498474185050625115214476, 5.67420337430965103043673963074, 6.50646523298619175836973026823, 7.09982448015459729271057825384, 8.673360246293950935582016928839, 9.557656775645400850261942594797, 9.900865126984285044446389180230