| L(s) = 1 | + (−1.67 + 0.448i)3-s − 0.896·5-s − i·7-s + (2.59 − 1.50i)9-s + 1.84i·13-s + (1.50 − 0.402i)15-s + 4.12i·17-s + 0.654·19-s + (0.448 + 1.67i)21-s − 7.12·23-s − 4.19·25-s + (−3.67 + 3.67i)27-s − 7.79·29-s + 0.896i·35-s − 10.6i·37-s + ⋯ |
| L(s) = 1 | + (−0.965 + 0.258i)3-s − 0.401·5-s − 0.377i·7-s + (0.865 − 0.500i)9-s + 0.512i·13-s + (0.387 − 0.103i)15-s + 1.00i·17-s + 0.150·19-s + (0.0978 + 0.365i)21-s − 1.48·23-s − 0.839·25-s + (−0.706 + 0.707i)27-s − 1.44·29-s + 0.151i·35-s − 1.75i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 - 0.292i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.956 - 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0326480 + 0.217993i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0326480 + 0.217993i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.67 - 0.448i)T \) |
| 7 | \( 1 + iT \) |
| good | 5 | \( 1 + 0.896T + 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 1.84iT - 13T^{2} \) |
| 17 | \( 1 - 4.12iT - 17T^{2} \) |
| 19 | \( 1 - 0.654T + 19T^{2} \) |
| 23 | \( 1 + 7.12T + 23T^{2} \) |
| 29 | \( 1 + 7.79T + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 10.6iT - 37T^{2} \) |
| 41 | \( 1 - 10.1iT - 41T^{2} \) |
| 43 | \( 1 - 1.19T + 43T^{2} \) |
| 47 | \( 1 + 9.59T + 47T^{2} \) |
| 53 | \( 1 + 8.24T + 53T^{2} \) |
| 59 | \( 1 - 6.89iT - 59T^{2} \) |
| 61 | \( 1 - 4.54iT - 61T^{2} \) |
| 67 | \( 1 - 5.49T + 67T^{2} \) |
| 71 | \( 1 - 4.12T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 13.3iT - 79T^{2} \) |
| 83 | \( 1 - 13.3iT - 83T^{2} \) |
| 89 | \( 1 + 13.7iT - 89T^{2} \) |
| 97 | \( 1 + 8.99T + 97T^{2} \) |
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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
−11.07315321878206461738434875512, −10.06560133324475624358190492278, −9.453446709422517233016835702336, −8.146906894269722684385562986448, −7.35225759834224660768387313078, −6.31254793686027730153747369478, −5.59852080415097180576293122194, −4.31538456145551234006297725094, −3.75201102225018579089186272708, −1.72102239180814825761060095540,
0.12996123021008500749477993652, 1.92716004911322060342483326951, 3.52996846413259460117574161091, 4.75509027620398015954465387762, 5.60346466826902836430261578389, 6.44836356625613118319979124335, 7.50271497150445896976137398062, 8.123770581389422679872434149331, 9.465547511041898865906455259209, 10.14647988250190226042605358728
