| L(s) = 1 | + i·3-s + 1.41·7-s − 9-s + 1.41i·13-s − 6·17-s − 6i·19-s + 1.41i·21-s − 8.48·23-s + 5·25-s − i·27-s − 8.48i·29-s + 1.41·31-s − 7.07i·37-s − 1.41·39-s + 6·41-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.534·7-s − 0.333·9-s + 0.392i·13-s − 1.45·17-s − 1.37i·19-s + 0.308i·21-s − 1.76·23-s + 25-s − 0.192i·27-s − 1.57i·29-s + 0.254·31-s − 1.16i·37-s − 0.226·39-s + 0.937·41-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
| \(L(1)\) |
\(\approx\) |
\(0.9364344353\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9364344353\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 5T^{2} \) |
| 7 | \( 1 - 1.41T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 1.41iT - 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 + 6iT - 19T^{2} \) |
| 23 | \( 1 + 8.48T + 23T^{2} \) |
| 29 | \( 1 + 8.48iT - 29T^{2} \) |
| 31 | \( 1 - 1.41T + 31T^{2} \) |
| 37 | \( 1 + 7.07iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 6iT - 43T^{2} \) |
| 47 | \( 1 + 8.48T + 47T^{2} \) |
| 53 | \( 1 + 8.48iT - 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 7.07iT - 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 8.48T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 1.41T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 12T + 97T^{2} \) |
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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
−8.438174708430518512099641407886, −8.026870753064438546281508331464, −6.87814974905949407889497664794, −6.32795152709152376496503148672, −5.31395498027439616614026610426, −4.47685898806043931163223850136, −4.07936923970635971950703734797, −2.73667264220938017123003260660, −1.97040003953426461633677681274, −0.28339864770998730113055690983,
1.32588051780252714806230688703, 2.16158226783656086960862977224, 3.24001953078518237532610773919, 4.25882852792449715463878995562, 5.07300400569847056263394025325, 5.98975169266293578725729303150, 6.61153965984447952708549617713, 7.44900327008433671629854517319, 8.222774951847127131800659742433, 8.607928245133438455818813721420
