| L(s) = 1 | + (0.0747 + 0.997i)2-s + (1.95 + 0.601i)3-s + (−0.988 + 0.149i)4-s + (0.958 − 0.889i)5-s + (−0.454 + 1.99i)6-s + (−2.43 − 1.02i)7-s + (−0.222 − 0.974i)8-s + (0.964 + 0.657i)9-s + (0.958 + 0.889i)10-s + (−2.30 + 1.57i)11-s + (−2.01 − 0.304i)12-s + (−0.810 − 0.390i)13-s + (0.843 − 2.50i)14-s + (2.40 − 1.15i)15-s + (0.955 − 0.294i)16-s + (0.719 + 1.83i)17-s + ⋯ |
| L(s) = 1 | + (0.0528 + 0.705i)2-s + (1.12 + 0.347i)3-s + (−0.494 + 0.0745i)4-s + (0.428 − 0.397i)5-s + (−0.185 + 0.812i)6-s + (−0.921 − 0.388i)7-s + (−0.0786 − 0.344i)8-s + (0.321 + 0.219i)9-s + (0.302 + 0.281i)10-s + (−0.694 + 0.473i)11-s + (−0.582 − 0.0878i)12-s + (−0.224 − 0.108i)13-s + (0.225 − 0.670i)14-s + (0.620 − 0.298i)15-s + (0.238 − 0.0736i)16-s + (0.174 + 0.444i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.612 - 0.790i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.13543 + 0.556175i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.13543 + 0.556175i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0747 - 0.997i)T \) |
| 7 | \( 1 + (2.43 + 1.02i)T \) |
| good | 3 | \( 1 + (-1.95 - 0.601i)T + (2.47 + 1.68i)T^{2} \) |
| 5 | \( 1 + (-0.958 + 0.889i)T + (0.373 - 4.98i)T^{2} \) |
| 11 | \( 1 + (2.30 - 1.57i)T + (4.01 - 10.2i)T^{2} \) |
| 13 | \( 1 + (0.810 + 0.390i)T + (8.10 + 10.1i)T^{2} \) |
| 17 | \( 1 + (-0.719 - 1.83i)T + (-12.4 + 11.5i)T^{2} \) |
| 19 | \( 1 + (-0.387 - 0.670i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.28 + 5.82i)T + (-16.8 - 15.6i)T^{2} \) |
| 29 | \( 1 + (-6.52 + 8.18i)T + (-6.45 - 28.2i)T^{2} \) |
| 31 | \( 1 + (4.21 - 7.30i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.65 - 0.551i)T + (35.3 + 10.9i)T^{2} \) |
| 41 | \( 1 + (1.26 + 5.55i)T + (-36.9 + 17.7i)T^{2} \) |
| 43 | \( 1 + (2.00 - 8.79i)T + (-38.7 - 18.6i)T^{2} \) |
| 47 | \( 1 + (-0.907 - 12.1i)T + (-46.4 + 7.00i)T^{2} \) |
| 53 | \( 1 + (-6.87 + 1.03i)T + (50.6 - 15.6i)T^{2} \) |
| 59 | \( 1 + (-3.85 - 3.57i)T + (4.40 + 58.8i)T^{2} \) |
| 61 | \( 1 + (12.1 + 1.83i)T + (58.2 + 17.9i)T^{2} \) |
| 67 | \( 1 + (-0.734 + 1.27i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3.74 + 4.69i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (-0.00237 + 0.0316i)T + (-72.1 - 10.8i)T^{2} \) |
| 79 | \( 1 + (6.94 + 12.0i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (9.26 - 4.46i)T + (51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (-2.43 - 1.65i)T + (32.5 + 82.8i)T^{2} \) |
| 97 | \( 1 - 9.86T + 97T^{2} \) |
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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
−14.16582538746063549218203534076, −13.27428570164243260633669487971, −12.50198067447417121603333079597, −10.35584555790047967036320506430, −9.540678383574186501015694779712, −8.604287389463686030641281597070, −7.49605537823183155598202755696, −6.09805512566990279835227889894, −4.50209321831877829616566175586, −2.96074271739291827723826869601,
2.44346085859508157698623224379, 3.33585878712522198921358511317, 5.52988871076165327509959531835, 7.15174290855096719889254692963, 8.519682148146022845120027846277, 9.436654633191023392513908393076, 10.40197486889344287259714107471, 11.75663885594855483568148975160, 13.03648283304788764267815955812, 13.56542654100203951933072929508
