| L(s) = 1 | + (−19.5 + 11.3i)2-s + (57.7 + 236. i)3-s + (255. − 443. i)4-s + (−2.30e3 − 1.33e3i)5-s + (−3.80e3 − 3.97e3i)6-s + (−3.51e3 − 6.08e3i)7-s + 1.15e4i·8-s + (−5.23e4 + 2.72e4i)9-s + 6.02e4·10-s + (2.13e4 − 1.23e4i)11-s + (1.19e5 + 3.48e4i)12-s + (2.70e5 − 4.68e5i)13-s + (1.37e5 + 7.95e4i)14-s + (1.81e5 − 6.21e5i)15-s + (−1.31e5 − 2.27e5i)16-s − 1.33e6i·17-s + ⋯ |
| L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.237 + 0.971i)3-s + (0.249 − 0.433i)4-s + (−0.737 − 0.425i)5-s + (−0.488 − 0.510i)6-s + (−0.209 − 0.362i)7-s + 0.353i·8-s + (−0.887 + 0.461i)9-s + 0.602·10-s + (0.132 − 0.0764i)11-s + (0.480 + 0.139i)12-s + (0.728 − 1.26i)13-s + (0.256 + 0.147i)14-s + (0.238 − 0.817i)15-s + (−0.125 − 0.216i)16-s − 0.943i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.608 + 0.793i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.715586 - 0.353042i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.715586 - 0.353042i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (19.5 - 11.3i)T \) |
| 3 | \( 1 + (-57.7 - 236. i)T \) |
| good | 5 | \( 1 + (2.30e3 + 1.33e3i)T + (4.88e6 + 8.45e6i)T^{2} \) |
| 7 | \( 1 + (3.51e3 + 6.08e3i)T + (-1.41e8 + 2.44e8i)T^{2} \) |
| 11 | \( 1 + (-2.13e4 + 1.23e4i)T + (1.29e10 - 2.24e10i)T^{2} \) |
| 13 | \( 1 + (-2.70e5 + 4.68e5i)T + (-6.89e10 - 1.19e11i)T^{2} \) |
| 17 | \( 1 + 1.33e6iT - 2.01e12T^{2} \) |
| 19 | \( 1 - 3.09e6T + 6.13e12T^{2} \) |
| 23 | \( 1 + (-2.15e6 - 1.24e6i)T + (2.07e13 + 3.58e13i)T^{2} \) |
| 29 | \( 1 + (2.67e7 - 1.54e7i)T + (2.10e14 - 3.64e14i)T^{2} \) |
| 31 | \( 1 + (-1.67e7 + 2.89e7i)T + (-4.09e14 - 7.09e14i)T^{2} \) |
| 37 | \( 1 + 3.54e7T + 4.80e15T^{2} \) |
| 41 | \( 1 + (1.05e8 + 6.10e7i)T + (6.71e15 + 1.16e16i)T^{2} \) |
| 43 | \( 1 + (1.18e8 + 2.05e8i)T + (-1.08e16 + 1.87e16i)T^{2} \) |
| 47 | \( 1 + (-5.31e7 + 3.06e7i)T + (2.62e16 - 4.55e16i)T^{2} \) |
| 53 | \( 1 - 3.59e8iT - 1.74e17T^{2} \) |
| 59 | \( 1 + (8.65e8 + 4.99e8i)T + (2.55e17 + 4.42e17i)T^{2} \) |
| 61 | \( 1 + (1.65e8 + 2.85e8i)T + (-3.56e17 + 6.17e17i)T^{2} \) |
| 67 | \( 1 + (9.42e8 - 1.63e9i)T + (-9.11e17 - 1.57e18i)T^{2} \) |
| 71 | \( 1 - 1.92e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 - 2.12e9T + 4.29e18T^{2} \) |
| 79 | \( 1 + (1.67e8 + 2.90e8i)T + (-4.73e18 + 8.19e18i)T^{2} \) |
| 83 | \( 1 + (-5.02e9 + 2.90e9i)T + (7.75e18 - 1.34e19i)T^{2} \) |
| 89 | \( 1 + 1.02e10iT - 3.11e19T^{2} \) |
| 97 | \( 1 + (-5.48e9 - 9.50e9i)T + (-3.68e19 + 6.38e19i)T^{2} \) |
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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
−15.99043417615301716687396368530, −15.31033934762495598283573441276, −13.68266491987183093575743699232, −11.60047655518057805341689061591, −10.26771763369224977950320858893, −8.930698387980429475022268847077, −7.64295526509202331965920368397, −5.32865880998141494207350638972, −3.47396082334519368466819144701, −0.44381490262993217299827613638,
1.54435693365863549505116885424, 3.38026199651728517675935662020, 6.51769486992374685524184672264, 7.83894131503074214544610437397, 9.173678048718372686647603869962, 11.23259114757426153247593920414, 12.13439720954650894861380422545, 13.63382701123202567601021169665, 15.18045414318985129610414619798, 16.68750241866665522610645090545
