| L(s) = 1 | + (−1.84 − 0.766i)2-s + (−2.58 − 1.49i)3-s + (2.82 + 2.83i)4-s + (3.40 + 5.89i)5-s + (3.62 + 4.73i)6-s + (−3.04 − 7.39i)8-s + (−0.0457 − 0.0792i)9-s + (−1.76 − 13.4i)10-s + (−12.4 − 7.17i)11-s + (−3.06 − 11.5i)12-s + 4.62·13-s − 20.3i·15-s + (−0.0560 + 15.9i)16-s + (−5.80 + 10.0i)17-s + (0.0237 + 0.181i)18-s + (−19.4 + 11.2i)19-s + ⋯ |
| L(s) = 1 | + (−0.923 − 0.383i)2-s + (−0.861 − 0.497i)3-s + (0.705 + 0.708i)4-s + (0.680 + 1.17i)5-s + (0.604 + 0.789i)6-s + (−0.380 − 0.924i)8-s + (−0.00508 − 0.00880i)9-s + (−0.176 − 1.34i)10-s + (−1.12 − 0.651i)11-s + (−0.255 − 0.961i)12-s + 0.355·13-s − 1.35i·15-s + (−0.00350 + 0.999i)16-s + (−0.341 + 0.591i)17-s + (0.00131 + 0.0100i)18-s + (−1.02 + 0.590i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.789 - 0.613i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.789 - 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0422605 + 0.123299i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0422605 + 0.123299i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.84 + 0.766i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + (2.58 + 1.49i)T + (4.5 + 7.79i)T^{2} \) |
| 5 | \( 1 + (-3.40 - 5.89i)T + (-12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (12.4 + 7.17i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 - 4.62T + 169T^{2} \) |
| 17 | \( 1 + (5.80 - 10.0i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (19.4 - 11.2i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (0.739 - 0.426i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 42.8T + 841T^{2} \) |
| 31 | \( 1 + (1.34 + 0.774i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (22.5 + 38.9i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + 36.6T + 1.68e3T^{2} \) |
| 43 | \( 1 - 27.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (36.5 - 21.0i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (21.7 - 37.5i)T + (-1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (46.7 + 26.9i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-7.97 - 13.8i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-79.6 - 45.9i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 16.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (4.79 - 8.30i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-49.3 + 28.4i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 14.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (50.0 + 86.7i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 68.2T + 9.40e3T^{2} \) |
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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
−12.53387085761898814216867955524, −11.16516756976932384304745553024, −10.89086581467101308546366767495, −10.00504869204678831295485170601, −8.686330286754226125636263126299, −7.52297421362407040404113664376, −6.44755279184345789811328154314, −5.83473851304326850965974079269, −3.36164411744223410308346426187, −1.96782440672006883221272870603,
0.10321796267630985755165542674, 2.03247981108756044349422873880, 4.90078036465015034527488299737, 5.37523709960078928826104057350, 6.61128425125678970255842620026, 8.019417115411264838666729518747, 8.970062507878484141660791611906, 9.892760392676307161409152250994, 10.67778340976670987841506420340, 11.57518713447773512616107745242
