L(s) = 1 | + (−0.305 + 1.14i)2-s + (2.30 + 3.98i)3-s + (2.25 + 1.30i)4-s + (−1.58 − 1.58i)5-s + (−5.25 + 1.40i)6-s + (−3.05 − 11.4i)7-s + (−5.51 + 5.51i)8-s + (−6.08 + 10.5i)9-s + (2.28 − 1.32i)10-s + (16.5 + 4.43i)11-s + 11.9i·12-s + (−9.22 − 9.15i)13-s + 13.9·14-s + (2.66 − 9.93i)15-s + (0.590 + 1.02i)16-s + (−2.58 − 1.49i)17-s + ⋯ |
L(s) = 1 | + (−0.152 + 0.570i)2-s + (0.767 + 1.32i)3-s + (0.563 + 0.325i)4-s + (−0.316 − 0.316i)5-s + (−0.875 + 0.234i)6-s + (−0.436 − 1.62i)7-s + (−0.689 + 0.689i)8-s + (−0.676 + 1.17i)9-s + (0.228 − 0.132i)10-s + (1.50 + 0.403i)11-s + 0.998i·12-s + (−0.709 − 0.704i)13-s + 0.997·14-s + (0.177 − 0.662i)15-s + (0.0369 + 0.0639i)16-s + (−0.151 − 0.0877i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0421 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0421 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.00234 + 1.04550i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.00234 + 1.04550i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (1.58 + 1.58i)T \) |
| 13 | \( 1 + (9.22 + 9.15i)T \) |
good | 2 | \( 1 + (0.305 - 1.14i)T + (-3.46 - 2i)T^{2} \) |
| 3 | \( 1 + (-2.30 - 3.98i)T + (-4.5 + 7.79i)T^{2} \) |
| 7 | \( 1 + (3.05 + 11.4i)T + (-42.4 + 24.5i)T^{2} \) |
| 11 | \( 1 + (-16.5 - 4.43i)T + (104. + 60.5i)T^{2} \) |
| 17 | \( 1 + (2.58 + 1.49i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (7.87 - 2.10i)T + (312. - 180.5i)T^{2} \) |
| 23 | \( 1 + (-12.9 + 7.47i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-6.17 - 10.7i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (22.8 + 22.8i)T + 961iT^{2} \) |
| 37 | \( 1 + (-44.9 - 12.0i)T + (1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + (-2.59 + 9.67i)T + (-1.45e3 - 840.5i)T^{2} \) |
| 43 | \( 1 + (42.2 + 24.3i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (55.7 - 55.7i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + 19.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + (2.26 + 8.43i)T + (-3.01e3 + 1.74e3i)T^{2} \) |
| 61 | \( 1 + (-32.1 + 55.6i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (1.56 - 5.84i)T + (-3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 + (55.5 - 14.8i)T + (4.36e3 - 2.52e3i)T^{2} \) |
| 73 | \( 1 + (25.0 - 25.0i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + 6.57T + 6.24e3T^{2} \) |
| 83 | \( 1 + (26.1 + 26.1i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (-17.3 - 4.63i)T + (6.85e3 + 3.96e3i)T^{2} \) |
| 97 | \( 1 + (-121. + 32.6i)T + (8.14e3 - 4.70e3i)T^{2} \) |
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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
−14.92600219872301776160426932748, −14.41650735008870895983108832110, −12.84096970434151301443201120317, −11.31505944826737557030239594113, −10.15996842263767682177866576191, −9.155227186755528370133575725466, −7.85603752789314801775495204703, −6.73452663080187450368030870634, −4.47695825593204629180128919061, −3.40101482640906084542656733795,
1.85531316974596871934119693953, 3.04096601593050246712445727391, 6.21715629671314051186593838239, 6.97550689533610971205354665775, 8.657531438142730567541258994381, 9.489157917996490543100479047806, 11.51308426864332724351338950300, 12.00840266929612033571733284234, 12.96530293868076388286730783766, 14.52511300175756062658262206028