| L(s) = 1 | + (−0.647 + 1.25i)2-s + (0.547 + 3.10i)3-s + (−1.16 − 1.62i)4-s + (1.46 − 1.23i)5-s + (−4.26 − 1.32i)6-s + (−1.58 − 0.917i)7-s + (2.79 − 0.408i)8-s + (−6.53 + 2.38i)9-s + (0.598 + 2.64i)10-s + (2.10 − 1.21i)11-s + (4.42 − 4.50i)12-s + (2.49 + 0.440i)13-s + (2.18 − 1.40i)14-s + (4.62 + 3.88i)15-s + (−1.29 + 3.78i)16-s + (0.818 + 0.297i)17-s + ⋯ |
| L(s) = 1 | + (−0.457 + 0.889i)2-s + (0.316 + 1.79i)3-s + (−0.581 − 0.813i)4-s + (0.655 − 0.550i)5-s + (−1.74 − 0.539i)6-s + (−0.600 − 0.346i)7-s + (0.989 − 0.144i)8-s + (−2.17 + 0.793i)9-s + (0.189 + 0.834i)10-s + (0.635 − 0.366i)11-s + (1.27 − 1.30i)12-s + (0.692 + 0.122i)13-s + (0.583 − 0.375i)14-s + (1.19 + 1.00i)15-s + (−0.324 + 0.945i)16-s + (0.198 + 0.0722i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.471 - 0.881i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.471 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.424850 + 0.709130i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.424850 + 0.709130i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.647 - 1.25i)T \) |
| 19 | \( 1 + (4.35 + 0.0369i)T \) |
| good | 3 | \( 1 + (-0.547 - 3.10i)T + (-2.81 + 1.02i)T^{2} \) |
| 5 | \( 1 + (-1.46 + 1.23i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (1.58 + 0.917i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.10 + 1.21i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.49 - 0.440i)T + (12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.818 - 0.297i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (-4.69 + 5.59i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (0.967 + 2.65i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (2.07 - 3.59i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 1.48iT - 37T^{2} \) |
| 41 | \( 1 + (1.43 - 0.252i)T + (38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (5.26 + 6.27i)T + (-7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-0.855 - 2.34i)T + (-36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (1.18 - 1.41i)T + (-9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (-6.43 - 2.34i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (8.58 + 7.20i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-2.59 + 0.945i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (6.12 - 5.13i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-2.02 - 11.4i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (0.908 + 5.15i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-8.19 - 4.73i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (6.18 + 1.09i)T + (83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (2.11 - 5.81i)T + (-74.3 - 62.3i)T^{2} \) |
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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
−15.05658319069423123965215352474, −14.21804885856013311934604605706, −13.18560008124770770783219728002, −10.93482571954125291318103856411, −10.10475389036696841854948933848, −9.128337959861014492535173195638, −8.586134161987371752176027341782, −6.40108666389388120924604500671, −5.15622694516213646056835737151, −3.88574622954708484491988873326,
1.72706262235962652507823787028, 3.10062790110769999984955532617, 6.16244273790260290711291257881, 7.20310896585995174042526527490, 8.510900167216755830000101975005, 9.545418539713628666420275110275, 11.07665069438099097500600927347, 12.13918578591544475662460179002, 13.04152721046768570251116842993, 13.62170720854260973236423994382
