L(s) = 1 | + (687. + 1.19e3i)5-s + (−3.74e4 − 2.40e4i)7-s + (−5.64e5 − 3.26e5i)11-s − 9.58e5i·13-s + (−3.39e6 + 5.87e6i)17-s + (1.15e7 − 6.65e6i)19-s + (3.61e7 − 2.08e7i)23-s + (2.34e7 − 4.06e7i)25-s − 1.22e8i·29-s + (−8.30e7 − 4.79e7i)31-s + (2.89e6 − 6.11e7i)35-s + (1.74e8 + 3.02e8i)37-s − 1.06e8·41-s + 8.40e8·43-s + (−2.37e8 − 4.11e8i)47-s + ⋯ |
L(s) = 1 | + (0.0984 + 0.170i)5-s + (−0.841 − 0.540i)7-s + (−1.05 − 0.610i)11-s − 0.715i·13-s + (−0.579 + 1.00i)17-s + (1.06 − 0.616i)19-s + (1.17 − 0.676i)23-s + (0.480 − 0.832i)25-s − 1.11i·29-s + (−0.521 − 0.300i)31-s + (0.00930 − 0.196i)35-s + (0.413 + 0.716i)37-s − 0.144·41-s + 0.871·43-s + (−0.151 − 0.261i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.938 - 0.345i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.938 - 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(\approx\) |
\(0.5005505605\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5005505605\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (3.74e4 + 2.40e4i)T \) |
good | 5 | \( 1 + (-687. - 1.19e3i)T + (-2.44e7 + 4.22e7i)T^{2} \) |
| 11 | \( 1 + (5.64e5 + 3.26e5i)T + (1.42e11 + 2.47e11i)T^{2} \) |
| 13 | \( 1 + 9.58e5iT - 1.79e12T^{2} \) |
| 17 | \( 1 + (3.39e6 - 5.87e6i)T + (-1.71e13 - 2.96e13i)T^{2} \) |
| 19 | \( 1 + (-1.15e7 + 6.65e6i)T + (5.82e13 - 1.00e14i)T^{2} \) |
| 23 | \( 1 + (-3.61e7 + 2.08e7i)T + (4.76e14 - 8.25e14i)T^{2} \) |
| 29 | \( 1 + 1.22e8iT - 1.22e16T^{2} \) |
| 31 | \( 1 + (8.30e7 + 4.79e7i)T + (1.27e16 + 2.20e16i)T^{2} \) |
| 37 | \( 1 + (-1.74e8 - 3.02e8i)T + (-8.89e16 + 1.54e17i)T^{2} \) |
| 41 | \( 1 + 1.06e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 8.40e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + (2.37e8 + 4.11e8i)T + (-1.23e18 + 2.14e18i)T^{2} \) |
| 53 | \( 1 + (1.78e9 + 1.03e9i)T + (4.63e18 + 8.02e18i)T^{2} \) |
| 59 | \( 1 + (-8.22e8 + 1.42e9i)T + (-1.50e19 - 2.61e19i)T^{2} \) |
| 61 | \( 1 + (7.13e9 - 4.11e9i)T + (2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (-8.67e9 + 1.50e10i)T + (-6.10e19 - 1.05e20i)T^{2} \) |
| 71 | \( 1 - 1.85e10iT - 2.31e20T^{2} \) |
| 73 | \( 1 + (1.63e10 + 9.46e9i)T + (1.56e20 + 2.71e20i)T^{2} \) |
| 79 | \( 1 + (-1.50e10 - 2.61e10i)T + (-3.73e20 + 6.47e20i)T^{2} \) |
| 83 | \( 1 + 7.61e9T + 1.28e21T^{2} \) |
| 89 | \( 1 + (2.98e10 + 5.17e10i)T + (-1.38e21 + 2.40e21i)T^{2} \) |
| 97 | \( 1 + 1.23e9iT - 7.15e21T^{2} \) |
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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
−9.718135511689840293871550769568, −8.570498114447451896062521995331, −7.63624994837259740923696718115, −6.60299732531440660024490950092, −5.70418910345134449646153304614, −4.53249396853006619725470762706, −3.25155544399399531936181842579, −2.56253122896638068041778419640, −0.875002464596110105024954765875, −0.11125480172324162932784151984,
1.28755338921934542029792511195, 2.52780771107463892641429864928, 3.37879189334282766660935489685, 4.87590317964551034008649985829, 5.56174465534961849400230360437, 6.89494574628627313096462270751, 7.55551557161319792762881947201, 9.119049376733573528821367224574, 9.387375173867757631964562219733, 10.60651187337452909885790351494