| L(s) = 1 | + (−64 − 110. i)2-s + (−8.19e3 + 1.41e4i)4-s + (5.53e4 + 9.58e4i)5-s + (5.50e5 − 2.10e6i)7-s + 2.09e6·8-s + (7.08e6 − 1.22e7i)10-s + (2.69e7 − 4.66e7i)11-s + 2.39e8·13-s + (−2.68e8 + 7.38e7i)14-s + (−1.34e8 − 2.32e8i)16-s + (1.29e9 − 2.23e9i)17-s + (−2.01e9 − 3.48e9i)19-s − 1.81e9·20-s − 6.90e9·22-s + (3.30e9 + 5.73e9i)23-s + ⋯ |
| L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.316 + 0.548i)5-s + (0.252 − 0.967i)7-s + 0.353·8-s + (0.224 − 0.388i)10-s + (0.417 − 0.722i)11-s + 1.05·13-s + (−0.681 + 0.187i)14-s + (−0.125 − 0.216i)16-s + (0.763 − 1.32i)17-s + (−0.516 − 0.894i)19-s − 0.316·20-s − 0.589·22-s + (0.202 + 0.351i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.746 + 0.665i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (-0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(2.151722532\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.151722532\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (64 + 110. i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-5.50e5 + 2.10e6i)T \) |
| good | 5 | \( 1 + (-5.53e4 - 9.58e4i)T + (-1.52e10 + 2.64e10i)T^{2} \) |
| 11 | \( 1 + (-2.69e7 + 4.66e7i)T + (-2.08e15 - 3.61e15i)T^{2} \) |
| 13 | \( 1 - 2.39e8T + 5.11e16T^{2} \) |
| 17 | \( 1 + (-1.29e9 + 2.23e9i)T + (-1.43e18 - 2.47e18i)T^{2} \) |
| 19 | \( 1 + (2.01e9 + 3.48e9i)T + (-7.59e18 + 1.31e19i)T^{2} \) |
| 23 | \( 1 + (-3.30e9 - 5.73e9i)T + (-1.33e20 + 2.30e20i)T^{2} \) |
| 29 | \( 1 + 5.84e9T + 8.62e21T^{2} \) |
| 31 | \( 1 + (-1.50e11 + 2.60e11i)T + (-1.17e22 - 2.03e22i)T^{2} \) |
| 37 | \( 1 + (-7.37e10 - 1.27e11i)T + (-1.66e23 + 2.88e23i)T^{2} \) |
| 41 | \( 1 + 7.07e11T + 1.55e24T^{2} \) |
| 43 | \( 1 + 1.37e12T + 3.17e24T^{2} \) |
| 47 | \( 1 + (-1.97e12 - 3.42e12i)T + (-6.03e24 + 1.04e25i)T^{2} \) |
| 53 | \( 1 + (-3.72e12 + 6.44e12i)T + (-3.65e25 - 6.33e25i)T^{2} \) |
| 59 | \( 1 + (1.81e13 - 3.14e13i)T + (-1.82e26 - 3.16e26i)T^{2} \) |
| 61 | \( 1 + (-1.22e13 - 2.12e13i)T + (-3.01e26 + 5.21e26i)T^{2} \) |
| 67 | \( 1 + (6.14e12 - 1.06e13i)T + (-1.23e27 - 2.13e27i)T^{2} \) |
| 71 | \( 1 - 3.63e13T + 5.87e27T^{2} \) |
| 73 | \( 1 + (-4.65e13 + 8.05e13i)T + (-4.45e27 - 7.71e27i)T^{2} \) |
| 79 | \( 1 + (1.41e14 + 2.44e14i)T + (-1.45e28 + 2.52e28i)T^{2} \) |
| 83 | \( 1 - 2.65e14T + 6.11e28T^{2} \) |
| 89 | \( 1 + (-2.36e14 - 4.09e14i)T + (-8.70e28 + 1.50e29i)T^{2} \) |
| 97 | \( 1 + 4.86e14T + 6.33e29T^{2} \) |
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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
−10.32847072917263717504244737382, −9.336388760204536868403403488835, −8.239887393552427499454660920201, −7.15853235771291465339839305424, −6.11702432212926738435135223670, −4.59795754208967836151612103190, −3.51145236614405816802038365177, −2.58120330018364674700657578447, −1.15928787615216336826412623715, −0.51420983332434580260105196504,
1.20974227930340175763677479600, 1.84174071464799417513956472151, 3.58270229112906232021835785020, 4.90223742878067543012547849890, 5.82073980701757769331022639679, 6.67523072068574618091131972458, 8.240318263479714807877052105076, 8.655452658233463384497099561836, 9.761834451594621073288977710597, 10.76119616314439565714031963097
