| L(s) = 1 | + i·3-s − i·7-s − 9-s + 6·11-s − 5i·13-s + 6i·17-s − 5·19-s + 21-s + 6i·23-s − i·27-s − 6·29-s + 31-s + 6i·33-s + 2i·37-s + 5·39-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.377i·7-s − 0.333·9-s + 1.80·11-s − 1.38i·13-s + 1.45i·17-s − 1.14·19-s + 0.218·21-s + 1.25i·23-s − 0.192i·27-s − 1.11·29-s + 0.179·31-s + 1.04i·33-s + 0.328i·37-s + 0.800·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.918517580\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.918517580\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + iT - 7T^{2} \) |
| 11 | \( 1 - 6T + 11T^{2} \) |
| 13 | \( 1 + 5iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 + 5T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + iT - 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 - 12iT - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 - 13T + 61T^{2} \) |
| 67 | \( 1 + 11iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 7iT - 97T^{2} \) |
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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
−8.452293957893836008865088767680, −7.79269256947370971533460886624, −6.92172840405867299956095063243, −6.07951121888303215833571830546, −5.65478699821192405326711161179, −4.50940263472266057808710524758, −3.82997973735432705378852402607, −3.39873202620584695147564440633, −2.01728678140894120673065218438, −0.998007505833307047385853231120,
0.61784650159165587218007995658, 1.84873406468707104252775814054, 2.42766151147122454065430182708, 3.75105380915317385580510289194, 4.30564465930205384807314079011, 5.25983896758101428045009510703, 6.22039407250702694528961799349, 6.85818785654417770486200675013, 7.05039447424064324140924532244, 8.327226487992534623057315491699
