| L(s) = 1 | + (−19.7 + 60.8i)2-s + (719. − 522. i)3-s + (−3.31e3 − 2.40e3i)4-s + (552. + 1.69e3i)5-s + (1.75e4 + 5.41e4i)6-s + (−8.51e4 − 6.18e4i)7-s + (2.12e5 − 1.54e5i)8-s + (−2.48e5 + 7.64e5i)9-s − 1.14e5·10-s + (3.42e6 + 4.77e6i)11-s − 3.64e6·12-s + (8.22e6 − 2.53e7i)13-s + (5.44e6 − 3.95e6i)14-s + (1.28e6 + 9.33e5i)15-s + (5.18e6 + 1.59e7i)16-s + (−3.16e7 − 9.74e7i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 + 0.672i)2-s + (0.569 − 0.413i)3-s + (−0.404 − 0.293i)4-s + (0.0158 + 0.0486i)5-s + (0.153 + 0.473i)6-s + (−0.273 − 0.198i)7-s + (0.286 − 0.207i)8-s + (−0.155 + 0.479i)9-s − 0.0361·10-s + (0.583 + 0.812i)11-s − 0.352·12-s + (0.472 − 1.45i)13-s + (0.193 − 0.140i)14-s + (0.0291 + 0.0211i)15-s + (0.0772 + 0.237i)16-s + (−0.318 − 0.979i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0402i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0402i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(2.027249369\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.027249369\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (19.7 - 60.8i)T \) |
| 11 | \( 1 + (-3.42e6 - 4.77e6i)T \) |
| good | 3 | \( 1 + (-719. + 522. i)T + (4.92e5 - 1.51e6i)T^{2} \) |
| 5 | \( 1 + (-552. - 1.69e3i)T + (-9.87e8 + 7.17e8i)T^{2} \) |
| 7 | \( 1 + (8.51e4 + 6.18e4i)T + (2.99e10 + 9.21e10i)T^{2} \) |
| 13 | \( 1 + (-8.22e6 + 2.53e7i)T + (-2.45e14 - 1.78e14i)T^{2} \) |
| 17 | \( 1 + (3.16e7 + 9.74e7i)T + (-8.01e15 + 5.82e15i)T^{2} \) |
| 19 | \( 1 + (-4.40e7 + 3.20e7i)T + (1.29e16 - 3.99e16i)T^{2} \) |
| 23 | \( 1 - 9.14e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + (-3.78e9 - 2.74e9i)T + (3.17e18 + 9.75e18i)T^{2} \) |
| 31 | \( 1 + (-1.34e9 + 4.14e9i)T + (-1.97e19 - 1.43e19i)T^{2} \) |
| 37 | \( 1 + (-2.72e9 - 1.98e9i)T + (7.52e19 + 2.31e20i)T^{2} \) |
| 41 | \( 1 + (-2.63e9 + 1.91e9i)T + (2.85e20 - 8.79e20i)T^{2} \) |
| 43 | \( 1 - 6.05e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + (2.84e10 - 2.06e10i)T + (1.68e21 - 5.19e21i)T^{2} \) |
| 53 | \( 1 + (2.39e10 - 7.38e10i)T + (-2.10e22 - 1.53e22i)T^{2} \) |
| 59 | \( 1 + (9.15e10 + 6.64e10i)T + (3.24e22 + 9.98e22i)T^{2} \) |
| 61 | \( 1 + (1.78e11 + 5.48e11i)T + (-1.30e23 + 9.51e22i)T^{2} \) |
| 67 | \( 1 + 7.67e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + (1.90e11 + 5.85e11i)T + (-9.42e23 + 6.84e23i)T^{2} \) |
| 73 | \( 1 + (-1.52e12 - 1.10e12i)T + (5.16e23 + 1.59e24i)T^{2} \) |
| 79 | \( 1 + (-4.48e11 + 1.37e12i)T + (-3.77e24 - 2.74e24i)T^{2} \) |
| 83 | \( 1 + (-7.45e11 - 2.29e12i)T + (-7.17e24 + 5.21e24i)T^{2} \) |
| 89 | \( 1 - 4.79e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + (2.13e12 - 6.55e12i)T + (-5.44e25 - 3.95e25i)T^{2} \) |
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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
−14.90726289253106370779950405669, −13.76863738155980670277024004380, −12.70436324352651836937322500258, −10.70527769858972518064981458084, −9.189828507477739012392787723468, −7.897281472698398458608879421274, −6.76000441406758759690371968863, −4.95490602552646822455619322540, −2.85734315644417783353699656363, −0.889971329973001985189416991795,
1.16012497596188126992244085130, 2.98711974598631572195916500061, 4.18237308742777150463239291117, 6.42155835536445711455485289044, 8.659665884846784806932728258107, 9.292677078109714219828709440239, 10.91028359292813876452942268613, 12.12571774058379971811847381082, 13.63172730683011490049932995487, 14.72306672672183686518989161272
