| L(s) = 1 | + (1 − 2i)5-s − 4i·7-s − 4i·13-s + 8·19-s + 4i·23-s + (−3 − 4i)25-s + 6·29-s + 8·31-s + (−8 − 4i)35-s − 4i·37-s − 6·41-s + 4i·43-s + 4i·47-s − 9·49-s − 12i·53-s + ⋯ |
| L(s) = 1 | + (0.447 − 0.894i)5-s − 1.51i·7-s − 1.10i·13-s + 1.83·19-s + 0.834i·23-s + (−0.600 − 0.800i)25-s + 1.11·29-s + 1.43·31-s + (−1.35 − 0.676i)35-s − 0.657i·37-s − 0.937·41-s + 0.609i·43-s + 0.583i·47-s − 1.28·49-s − 1.64i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.023168686\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.023168686\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1 + 2i)T \) |
| good | 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 8T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 4iT - 47T^{2} \) |
| 53 | \( 1 + 12iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 + 16T + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−8.365461418906253450800685106449, −7.83100285666323959724319695855, −7.14444368633372476856315203201, −6.20968661763707569030154193342, −5.28559974798868062265604710560, −4.74556533617142774785665008884, −3.75011043059342512679887158877, −2.92244067888758713257899134243, −1.35105777730234375396037589766, −0.71024289263720073803820861234,
1.51186729898671539299119058623, 2.62640613061990272688512021214, 3.02783462273235705876956197288, 4.38720648724227894710446407446, 5.30522118372809617491090806519, 6.00846662563900117861344616017, 6.67493755007643118678962109518, 7.36550396291301247274156863202, 8.486196106752036718806154990659, 8.926272091968759069212248150115
