| L(s) = 1 | − 1.73i·3-s + 8.89·5-s + 2.82i·7-s − 2.99·9-s − 18.2i·11-s − 5.79·13-s − 15.4i·15-s − 21.5·17-s − 18.2i·19-s + 4.89·21-s − 33.3i·23-s + 54.1·25-s + 5.19i·27-s − 4.49·29-s − 2.25i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 1.77·5-s + 0.404i·7-s − 0.333·9-s − 1.65i·11-s − 0.445·13-s − 1.02i·15-s − 1.27·17-s − 0.960i·19-s + 0.233·21-s − 1.45i·23-s + 2.16·25-s + 0.192i·27-s − 0.154·29-s − 0.0728i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.248617576\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.248617576\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| good | 5 | \( 1 - 8.89T + 25T^{2} \) |
| 7 | \( 1 - 2.82iT - 49T^{2} \) |
| 11 | \( 1 + 18.2iT - 121T^{2} \) |
| 13 | \( 1 + 5.79T + 169T^{2} \) |
| 17 | \( 1 + 21.5T + 289T^{2} \) |
| 19 | \( 1 + 18.2iT - 361T^{2} \) |
| 23 | \( 1 + 33.3iT - 529T^{2} \) |
| 29 | \( 1 + 4.49T + 841T^{2} \) |
| 31 | \( 1 + 2.25iT - 961T^{2} \) |
| 37 | \( 1 - 43.1T + 1.36e3T^{2} \) |
| 41 | \( 1 - 1.59T + 1.68e3T^{2} \) |
| 43 | \( 1 - 63.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 72.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 70.2T + 2.80e3T^{2} \) |
| 59 | \( 1 - 34.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 63.5T + 3.72e3T^{2} \) |
| 67 | \( 1 - 3.24iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 68.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 10T + 5.32e3T^{2} \) |
| 79 | \( 1 + 35.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 42.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 5.19T + 7.92e3T^{2} \) |
| 97 | \( 1 + 26.8T + 9.40e3T^{2} \) |
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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
−9.850974781507147492900796252618, −8.849172767821721302546055517540, −8.594796305994864692943400177031, −7.04768040641855914320018436570, −6.22566617728812207218556579004, −5.76435413505731685460235336730, −4.67803537631472615827829871505, −2.80233275093591062667292651094, −2.23417411110678447708453411007, −0.74540668008873571396194302001,
1.66508507299746835591655952741, 2.47842774047467675162809256834, 4.07611535242803702264794815310, 4.98967070468349598792328807662, 5.81459409450596204527550246736, 6.76426855802276683061480539360, 7.64509418240221267663008476015, 9.113024497850087829296000637594, 9.537062329943614182275638268549, 10.18015552297063689182943219883
