| L(s) = 1 | − i·3-s − 2.14·5-s + 1.11i·7-s − 9-s + 2.16i·11-s + 2.31i·13-s + 2.14i·15-s − 5.16i·17-s − 6.83i·19-s + 1.11·21-s + 2.59·23-s − 0.390·25-s + i·27-s − 0.866i·29-s − 5.09·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.960·5-s + 0.422i·7-s − 0.333·9-s + 0.652i·11-s + 0.641i·13-s + 0.554i·15-s − 1.25i·17-s − 1.56i·19-s + 0.243·21-s + 0.540·23-s − 0.0780·25-s + 0.192i·27-s − 0.160i·29-s − 0.915·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.677 - 0.735i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.677 - 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.015616901\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.015616901\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 41 | \( 1 + (-4.70 - 4.33i)T \) |
| good | 5 | \( 1 + 2.14T + 5T^{2} \) |
| 7 | \( 1 - 1.11iT - 7T^{2} \) |
| 11 | \( 1 - 2.16iT - 11T^{2} \) |
| 13 | \( 1 - 2.31iT - 13T^{2} \) |
| 17 | \( 1 + 5.16iT - 17T^{2} \) |
| 19 | \( 1 + 6.83iT - 19T^{2} \) |
| 23 | \( 1 - 2.59T + 23T^{2} \) |
| 29 | \( 1 + 0.866iT - 29T^{2} \) |
| 31 | \( 1 + 5.09T + 31T^{2} \) |
| 37 | \( 1 + 7.90T + 37T^{2} \) |
| 43 | \( 1 - 6.47T + 43T^{2} \) |
| 47 | \( 1 - 7.81iT - 47T^{2} \) |
| 53 | \( 1 - 0.106iT - 53T^{2} \) |
| 59 | \( 1 + 6.81T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 - 3.90iT - 67T^{2} \) |
| 71 | \( 1 - 11.2iT - 71T^{2} \) |
| 73 | \( 1 - 1.01T + 73T^{2} \) |
| 79 | \( 1 + 6.28iT - 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 - 17.8iT - 89T^{2} \) |
| 97 | \( 1 - 6.39iT - 97T^{2} \) |
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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
−8.607770178847148530728873304960, −7.56429831889348815453340437819, −7.22980206506503793329208318074, −6.62733696311291697809040463181, −5.52005702723678880342986619896, −4.77508324628639854275360097231, −4.04031500413132396822486024574, −2.92069688805210849750639881250, −2.22691079606322851453388962886, −0.863867154874662731032831054541,
0.38516713434384241423146317683, 1.78269648300715495657662022742, 3.27208656068711791508963569017, 3.72155962851439087131657722831, 4.31967815293033815879450453701, 5.51453566281782906773973098329, 5.90329810955072528659174107660, 7.07590377615422807865807258610, 7.73186843067112965991916532623, 8.392812276582323801457406841662
