| L(s) = 1 | − i·3-s + 3.86i·5-s + 2.44·7-s − 9-s − 1.46i·11-s − 4.24i·13-s + 3.86·15-s − 3.46·17-s − 7.46i·19-s − 2.44i·21-s + 2.82·23-s − 9.92·25-s + i·27-s − 8.76i·29-s − 7.34·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 1.72i·5-s + 0.925·7-s − 0.333·9-s − 0.441i·11-s − 1.17i·13-s + 0.997·15-s − 0.840·17-s − 1.71i·19-s − 0.534i·21-s + 0.589·23-s − 1.98·25-s + 0.192i·27-s − 1.62i·29-s − 1.31·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.382195952\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.382195952\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| good | 5 | \( 1 - 3.86iT - 5T^{2} \) |
| 7 | \( 1 - 2.44T + 7T^{2} \) |
| 11 | \( 1 + 1.46iT - 11T^{2} \) |
| 13 | \( 1 + 4.24iT - 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 7.46iT - 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 + 8.76iT - 29T^{2} \) |
| 31 | \( 1 + 7.34T + 31T^{2} \) |
| 37 | \( 1 - 0.656iT - 37T^{2} \) |
| 41 | \( 1 + 4.53T + 41T^{2} \) |
| 43 | \( 1 + 3.46iT - 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 + 4.62iT - 53T^{2} \) |
| 59 | \( 1 - 13.8iT - 59T^{2} \) |
| 61 | \( 1 - 0.656iT - 61T^{2} \) |
| 67 | \( 1 + 14.9iT - 67T^{2} \) |
| 71 | \( 1 - 6.41T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 - 2.44T + 79T^{2} \) |
| 83 | \( 1 + 5.46iT - 83T^{2} \) |
| 89 | \( 1 - 4.92T + 89T^{2} \) |
| 97 | \( 1 - 1.07T + 97T^{2} \) |
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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
−8.363254826738400846498747631345, −7.57095658248463919495202165156, −7.12406572304374558926409220540, −6.38292795152665325907287114412, −5.63603788010586230285383393175, −4.69703474780562610929674633733, −3.49493296933399153732717122050, −2.72733274163521646050720287110, −2.04575131314617586948965209839, −0.42095362951038571144898499439,
1.38122507721193970393332904551, 1.94488714456275599405769581445, 3.69510030581384917673974301310, 4.34846210839511482743821797433, 5.00785357353458193321981606038, 5.45907514504624557657809841306, 6.60211658452343353737335881088, 7.61623572918062612715414094273, 8.361256221726515512080532372627, 8.946444807599711756058062054260
