| L(s) = 1 | + (−1.77 − 0.921i)2-s + (2.30 + 3.27i)4-s + 1.07·5-s + 7.21·7-s + (−1.07 − 7.92i)8-s + (−1.90 − 0.990i)10-s + 16.3·11-s − 21.6i·13-s + (−12.8 − 6.64i)14-s + (−5.39 + 15.0i)16-s + 18.9i·17-s + 17.0i·19-s + (2.47 + 3.51i)20-s + (−29.0 − 15.0i)22-s + 1.11i·23-s + ⋯ |
| L(s) = 1 | + (−0.887 − 0.460i)2-s + (0.575 + 0.817i)4-s + 0.214·5-s + 1.03·7-s + (−0.134 − 0.990i)8-s + (−0.190 − 0.0990i)10-s + 1.48·11-s − 1.66i·13-s + (−0.914 − 0.474i)14-s + (−0.337 + 0.941i)16-s + 1.11i·17-s + 0.897i·19-s + (0.123 + 0.175i)20-s + (−1.31 − 0.684i)22-s + 0.0485i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 + 0.462i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.886 + 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.949131 - 0.232620i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.949131 - 0.232620i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.77 + 0.921i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 1.07T + 25T^{2} \) |
| 7 | \( 1 - 7.21T + 49T^{2} \) |
| 11 | \( 1 - 16.3T + 121T^{2} \) |
| 13 | \( 1 + 21.6iT - 169T^{2} \) |
| 17 | \( 1 - 18.9iT - 289T^{2} \) |
| 19 | \( 1 - 17.0iT - 361T^{2} \) |
| 23 | \( 1 - 1.11iT - 529T^{2} \) |
| 29 | \( 1 + 29.4T + 841T^{2} \) |
| 31 | \( 1 - 5.63T + 961T^{2} \) |
| 37 | \( 1 + 17.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 27.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 52.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 64.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 35.9T + 2.80e3T^{2} \) |
| 59 | \( 1 + 56.8T + 3.48e3T^{2} \) |
| 61 | \( 1 - 69.3iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 69.3iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 98.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 37.6T + 5.32e3T^{2} \) |
| 79 | \( 1 + 127.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 7.75T + 6.88e3T^{2} \) |
| 89 | \( 1 - 76.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 4.84T + 9.40e3T^{2} \) |
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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
−14.45688311605044781899601447711, −12.92386923074103123527623277714, −11.88788929158869683045588654494, −10.88392725304479128027303510678, −9.853323764428402412505197572393, −8.543200995009783118711686453401, −7.66109993310817599311707366898, −5.96115217893513013603737199003, −3.77811778538907013750361112231, −1.59190263552323676275955853787,
1.71811595376210578469323126760, 4.66953098928050755540552697235, 6.37759977511157285116304916054, 7.43691227224612107022386888193, 8.928669093586210443476644492570, 9.541908843228140794866798269016, 11.32210784688047764159412499684, 11.66866378400644276419247309939, 13.94216684839076759412123280051, 14.43051492139237569288541420549
