| L(s) = 1 | + (−2.82 − 4.89i)2-s + (27.6 + 15.9i)3-s + (−15.9 + 27.7i)4-s + (111. − 64.5i)5-s − 180. i·6-s + (298. − 168. i)7-s + 181.·8-s + (143. + 248. i)9-s + (−632. − 365. i)10-s + (−1.18e3 + 2.05e3i)11-s + (−883. + 510. i)12-s − 820. i·13-s + (−1.67e3 − 986. i)14-s + 4.11e3·15-s + (−512. − 886. i)16-s + (−2.28e3 − 1.31e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 − 0.612i)2-s + (1.02 + 0.590i)3-s + (−0.249 + 0.433i)4-s + (0.894 − 0.516i)5-s − 0.834i·6-s + (0.870 − 0.491i)7-s + 0.353·8-s + (0.197 + 0.341i)9-s + (−0.632 − 0.365i)10-s + (−0.893 + 1.54i)11-s + (−0.511 + 0.295i)12-s − 0.373i·13-s + (−0.608 − 0.359i)14-s + 1.21·15-s + (−0.125 − 0.216i)16-s + (−0.464 − 0.268i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 + 0.395i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.918 + 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.67068 - 0.344604i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.67068 - 0.344604i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.82 + 4.89i)T \) |
| 7 | \( 1 + (-298. + 168. i)T \) |
| good | 3 | \( 1 + (-27.6 - 15.9i)T + (364.5 + 631. i)T^{2} \) |
| 5 | \( 1 + (-111. + 64.5i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 11 | \( 1 + (1.18e3 - 2.05e3i)T + (-8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 + 820. iT - 4.82e6T^{2} \) |
| 17 | \( 1 + (2.28e3 + 1.31e3i)T + (1.20e7 + 2.09e7i)T^{2} \) |
| 19 | \( 1 + (4.81e3 - 2.77e3i)T + (2.35e7 - 4.07e7i)T^{2} \) |
| 23 | \( 1 + (5.69e3 + 9.86e3i)T + (-7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 + 3.28e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + (-4.06e4 - 2.34e4i)T + (4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + (1.08e4 + 1.88e4i)T + (-1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 - 8.58e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 1.13e5T + 6.32e9T^{2} \) |
| 47 | \( 1 + (-3.33e4 + 1.92e4i)T + (5.38e9 - 9.33e9i)T^{2} \) |
| 53 | \( 1 + (1.66e4 - 2.87e4i)T + (-1.10e10 - 1.91e10i)T^{2} \) |
| 59 | \( 1 + (2.89e5 + 1.67e5i)T + (2.10e10 + 3.65e10i)T^{2} \) |
| 61 | \( 1 + (1.45e5 - 8.42e4i)T + (2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-1.61e5 + 2.79e5i)T + (-4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 - 3.25e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + (-9.34e4 - 5.39e4i)T + (7.56e10 + 1.31e11i)T^{2} \) |
| 79 | \( 1 + (-1.69e5 - 2.93e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 - 2.24e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + (-3.31e5 + 1.91e5i)T + (2.48e11 - 4.30e11i)T^{2} \) |
| 97 | \( 1 + 3.26e4iT - 8.32e11T^{2} \) |
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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
−18.11631055464896800518576117860, −17.19243803602800786788991417009, −15.28666507082893762196115681696, −14.00376561475919706654880481374, −12.66959283856143158294393175668, −10.48147323341610829278648470247, −9.423402513786970821001421720538, −7.997149627071705271107016265200, −4.57558884213832782254869380218, −2.13848555479060199688344947858,
2.25381316288422032692063158545, 5.88485815721681064297071919201, 7.85178318415484875848240046217, 8.949229191217516028065576902399, 10.90936030177533950974858395467, 13.47322450203172140901478619010, 14.11767140598336274272720521025, 15.43827650375374445264122887774, 17.26999473335090918871288365879, 18.48866656504988239263301911575
