| L(s) = 1 | + (64 − 110. i)2-s + (−1.32e3 − 2.29e3i)3-s + (−8.19e3 − 1.41e4i)4-s + (5.62e4 − 9.73e4i)5-s − 3.39e5·6-s + (8.75e5 − 1.99e6i)7-s − 2.09e6·8-s + (3.66e6 − 6.34e6i)9-s + (−7.19e6 − 1.24e7i)10-s + (−8.67e6 − 1.50e7i)11-s + (−2.17e7 + 3.76e7i)12-s − 1.47e8·13-s + (−1.65e8 − 2.24e8i)14-s − 2.98e8·15-s + (−1.34e8 + 2.32e8i)16-s + (4.40e8 + 7.62e8i)17-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.349 − 0.606i)3-s + (−0.249 − 0.433i)4-s + (0.321 − 0.557i)5-s − 0.494·6-s + (0.401 − 0.915i)7-s − 0.353·8-s + (0.255 − 0.441i)9-s + (−0.227 − 0.394i)10-s + (−0.134 − 0.232i)11-s + (−0.174 + 0.303i)12-s − 0.650·13-s + (−0.418 − 0.569i)14-s − 0.450·15-s + (−0.125 + 0.216i)16-s + (0.260 + 0.450i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(0.282065 + 1.59501i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.282065 + 1.59501i\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-64 + 110. i)T \) |
| 7 | \( 1 + (-8.75e5 + 1.99e6i)T \) |
| good | 3 | \( 1 + (1.32e3 + 2.29e3i)T + (-7.17e6 + 1.24e7i)T^{2} \) |
| 5 | \( 1 + (-5.62e4 + 9.73e4i)T + (-1.52e10 - 2.64e10i)T^{2} \) |
| 11 | \( 1 + (8.67e6 + 1.50e7i)T + (-2.08e15 + 3.61e15i)T^{2} \) |
| 13 | \( 1 + 1.47e8T + 5.11e16T^{2} \) |
| 17 | \( 1 + (-4.40e8 - 7.62e8i)T + (-1.43e18 + 2.47e18i)T^{2} \) |
| 19 | \( 1 + (5.31e8 - 9.20e8i)T + (-7.59e18 - 1.31e19i)T^{2} \) |
| 23 | \( 1 + (3.90e9 - 6.77e9i)T + (-1.33e20 - 2.30e20i)T^{2} \) |
| 29 | \( 1 + 1.50e11T + 8.62e21T^{2} \) |
| 31 | \( 1 + (3.78e10 + 6.55e10i)T + (-1.17e22 + 2.03e22i)T^{2} \) |
| 37 | \( 1 + (-1.65e11 + 2.85e11i)T + (-1.66e23 - 2.88e23i)T^{2} \) |
| 41 | \( 1 - 1.73e12T + 1.55e24T^{2} \) |
| 43 | \( 1 - 2.25e12T + 3.17e24T^{2} \) |
| 47 | \( 1 + (-2.29e12 + 3.97e12i)T + (-6.03e24 - 1.04e25i)T^{2} \) |
| 53 | \( 1 + (4.14e12 + 7.18e12i)T + (-3.65e25 + 6.33e25i)T^{2} \) |
| 59 | \( 1 + (-2.16e12 - 3.74e12i)T + (-1.82e26 + 3.16e26i)T^{2} \) |
| 61 | \( 1 + (7.31e12 - 1.26e13i)T + (-3.01e26 - 5.21e26i)T^{2} \) |
| 67 | \( 1 + (3.20e13 + 5.55e13i)T + (-1.23e27 + 2.13e27i)T^{2} \) |
| 71 | \( 1 + 3.29e13T + 5.87e27T^{2} \) |
| 73 | \( 1 + (-6.70e13 - 1.16e14i)T + (-4.45e27 + 7.71e27i)T^{2} \) |
| 79 | \( 1 + (-9.06e13 + 1.57e14i)T + (-1.45e28 - 2.52e28i)T^{2} \) |
| 83 | \( 1 - 5.42e13T + 6.11e28T^{2} \) |
| 89 | \( 1 + (-1.39e13 + 2.41e13i)T + (-8.70e28 - 1.50e29i)T^{2} \) |
| 97 | \( 1 + 1.18e15T + 6.33e29T^{2} \) |
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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
−14.76698888459767839947981494270, −13.35754695015538011702222434964, −12.44120951735387779352961697312, −11.02743894508556004241986668619, −9.498146869201157790508902985825, −7.45239733454708550140114596293, −5.65079398613928453633841689564, −3.97287218248102872454515459513, −1.72053120627976397594924362996, −0.54510936242542417610563553763,
2.46445866998570038587400457583, 4.59892844657501833706526776331, 5.80780716743319151878826827533, 7.56170593969276197798507495013, 9.413773362218619802153078707502, 10.95098342161768068462694856800, 12.53467758334276180587287977672, 14.23797600806528205566533760261, 15.26608253511675072210928180334, 16.39487133830358259277500800198
