| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.731 + 0.681i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.731 + 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.80103 - 0.709206i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.80103 - 0.709206i\) |
| \(L(2)\) |
|
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
−14.10268910529873508597423497921, −13.27813542830240250225271317475, −11.99359241789245809327371754452, −11.17989968410484277371099544735, −10.15116240865332117080781399474, −8.426318881515651251187249537700, −6.95522034780958778527603824466, −5.33400493619717620391159653240, −4.18697262229911461909533270410, −2.13228612868530461579985154647,
2.86089602535425140732862029318, 4.76502272990261567986439284145, 5.72629301603921812389749145900, 7.68462728144602563032557190352, 8.096084656006209415243573700560, 10.32811933217002803007999983307, 11.41133043111341071539078619148, 12.53900820133915010660471393633, 13.49730137983618472049311452386, 14.55400363914638221353080872422
