| L(s) = 1 | − 1.83i·2-s − 1.36·4-s − 3.80·5-s − 1.16i·8-s + 6.98i·10-s + 1.16i·11-s + 5.54i·13-s − 4.86·16-s + 3.17·17-s − 0.631i·19-s + 5.19·20-s + 2.13·22-s + 1.76i·23-s + 9.50·25-s + 10.1·26-s + ⋯ |
| L(s) = 1 | − 1.29i·2-s − 0.682·4-s − 1.70·5-s − 0.412i·8-s + 2.20i·10-s + 0.351i·11-s + 1.53i·13-s − 1.21·16-s + 0.770·17-s − 0.144i·19-s + 1.16·20-s + 0.455·22-s + 0.367i·23-s + 1.90·25-s + 1.99·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.024812126\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.024812126\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 1.83iT - 2T^{2} \) |
| 5 | \( 1 + 3.80T + 5T^{2} \) |
| 11 | \( 1 - 1.16iT - 11T^{2} \) |
| 13 | \( 1 - 5.54iT - 13T^{2} \) |
| 17 | \( 1 - 3.17T + 17T^{2} \) |
| 19 | \( 1 + 0.631iT - 19T^{2} \) |
| 23 | \( 1 - 1.76iT - 23T^{2} \) |
| 29 | \( 1 + 4.83iT - 29T^{2} \) |
| 31 | \( 1 - 4.27iT - 31T^{2} \) |
| 37 | \( 1 - 4.23T + 37T^{2} \) |
| 41 | \( 1 - 4.56T + 41T^{2} \) |
| 43 | \( 1 - 7.23T + 43T^{2} \) |
| 47 | \( 1 + 2.54T + 47T^{2} \) |
| 53 | \( 1 - 0.0724iT - 53T^{2} \) |
| 59 | \( 1 + 6.98T + 59T^{2} \) |
| 61 | \( 1 - 8.08iT - 61T^{2} \) |
| 67 | \( 1 - 5.09T + 67T^{2} \) |
| 71 | \( 1 - 4.76iT - 71T^{2} \) |
| 73 | \( 1 + 5.59iT - 73T^{2} \) |
| 79 | \( 1 - 17.0T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 - 11.5T + 89T^{2} \) |
| 97 | \( 1 - 0.688iT - 97T^{2} \) |
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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
−9.582595764841833931501094585746, −9.011053814124749980191044687375, −7.900261772619804341337886571194, −7.28051053922899935560748371366, −6.39267926055219585898076523317, −4.76010202162909877495256207836, −4.09369647882330595545681902357, −3.45439140550879819626301373808, −2.32605997206898775737152561212, −0.980921288260781301466566197420,
0.55936531763959491094071632986, 2.87855309289083544190784370148, 3.80169900344041070148985086700, 4.84315641366972731604292843634, 5.64457648830643036425349376821, 6.53881423522370949017263308870, 7.58630422203210480327402793900, 7.84322991035168180047030148981, 8.401812450673517555999448415969, 9.410926946753345201984868739879
