| L(s) = 1 | + (108. + 187. i)2-s + (−2.79e3 + 4.83e3i)3-s + (−7.03e3 + 1.21e4i)4-s + (1.39e5 + 2.41e5i)5-s − 1.20e6·6-s + (−2.17e6 − 1.16e5i)7-s + 4.04e6·8-s + (−8.40e6 − 1.45e7i)9-s + (−3.01e7 + 5.22e7i)10-s + (5.06e7 − 8.77e7i)11-s + (−3.92e7 − 6.80e7i)12-s − 1.11e8·13-s + (−2.13e8 − 4.20e8i)14-s − 1.55e9·15-s + (6.68e8 + 1.15e9i)16-s + (−6.15e8 + 1.06e9i)17-s + ⋯ |
| L(s) = 1 | + (0.597 + 1.03i)2-s + (−0.736 + 1.27i)3-s + (−0.214 + 0.371i)4-s + (0.797 + 1.38i)5-s − 1.76·6-s + (−0.998 − 0.0532i)7-s + 0.682·8-s + (−0.585 − 1.01i)9-s + (−0.953 + 1.65i)10-s + (0.783 − 1.35i)11-s + (−0.316 − 0.547i)12-s − 0.492·13-s + (−0.541 − 1.06i)14-s − 2.35·15-s + (0.622 + 1.07i)16-s + (−0.363 + 0.630i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.918 + 0.395i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (-0.918 + 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(0.379841 - 1.84094i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.379841 - 1.84094i\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (2.17e6 + 1.16e5i)T \) |
| good | 2 | \( 1 + (-108. - 187. i)T + (-1.63e4 + 2.83e4i)T^{2} \) |
| 3 | \( 1 + (2.79e3 - 4.83e3i)T + (-7.17e6 - 1.24e7i)T^{2} \) |
| 5 | \( 1 + (-1.39e5 - 2.41e5i)T + (-1.52e10 + 2.64e10i)T^{2} \) |
| 11 | \( 1 + (-5.06e7 + 8.77e7i)T + (-2.08e15 - 3.61e15i)T^{2} \) |
| 13 | \( 1 + 1.11e8T + 5.11e16T^{2} \) |
| 17 | \( 1 + (6.15e8 - 1.06e9i)T + (-1.43e18 - 2.47e18i)T^{2} \) |
| 19 | \( 1 + (-1.52e9 - 2.63e9i)T + (-7.59e18 + 1.31e19i)T^{2} \) |
| 23 | \( 1 + (-4.89e9 - 8.48e9i)T + (-1.33e20 + 2.30e20i)T^{2} \) |
| 29 | \( 1 - 3.38e10T + 8.62e21T^{2} \) |
| 31 | \( 1 + (-1.39e10 + 2.40e10i)T + (-1.17e22 - 2.03e22i)T^{2} \) |
| 37 | \( 1 + (5.69e10 + 9.86e10i)T + (-1.66e23 + 2.88e23i)T^{2} \) |
| 41 | \( 1 - 6.06e11T + 1.55e24T^{2} \) |
| 43 | \( 1 + 2.11e12T + 3.17e24T^{2} \) |
| 47 | \( 1 + (-1.60e10 - 2.77e10i)T + (-6.03e24 + 1.04e25i)T^{2} \) |
| 53 | \( 1 + (4.01e12 - 6.95e12i)T + (-3.65e25 - 6.33e25i)T^{2} \) |
| 59 | \( 1 + (-1.31e13 + 2.27e13i)T + (-1.82e26 - 3.16e26i)T^{2} \) |
| 61 | \( 1 + (-1.03e13 - 1.79e13i)T + (-3.01e26 + 5.21e26i)T^{2} \) |
| 67 | \( 1 + (-9.83e11 + 1.70e12i)T + (-1.23e27 - 2.13e27i)T^{2} \) |
| 71 | \( 1 - 1.02e14T + 5.87e27T^{2} \) |
| 73 | \( 1 + (3.47e13 - 6.02e13i)T + (-4.45e27 - 7.71e27i)T^{2} \) |
| 79 | \( 1 + (-1.90e12 - 3.29e12i)T + (-1.45e28 + 2.52e28i)T^{2} \) |
| 83 | \( 1 - 4.86e14T + 6.11e28T^{2} \) |
| 89 | \( 1 + (-2.60e13 - 4.51e13i)T + (-8.70e28 + 1.50e29i)T^{2} \) |
| 97 | \( 1 + 8.40e14T + 6.33e29T^{2} \) |
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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
−19.21791752550365625115455122008, −17.16197423516069559825563553609, −16.18469263916544797371634745680, −14.95946159471099972198821297860, −13.80308084532947724670754206811, −10.95934937292306815402936140281, −9.862679606579320147974506542970, −6.56234343072448804346376619990, −5.71258557645362363502095195958, −3.58956685151799734530498726637,
0.830002356462404758286203576164, 2.10020027889093252803569105173, 4.89692236285983673922688965397, 6.85965802619509759634867772679, 9.610231306514195802848639689849, 11.96733707266200766960496339470, 12.65755708607646214828540516652, 13.46767148664335999420408298301, 16.64170613679798677524861737091, 17.68863723668098307606904937056
