| L(s) = 1 | + (−627. − 362. i)2-s + (2.81e4 − 5.18e4i)3-s + (2.62e5 + 4.54e5i)4-s + (1.35e7 − 7.83e6i)5-s + (−3.64e7 + 2.23e7i)6-s + (2.49e8 − 4.31e8i)7-s − 3.79e8i·8-s + (−1.89e9 − 2.92e9i)9-s − 1.13e10·10-s + (−4.34e9 − 2.50e9i)11-s + (3.09e10 − 8.08e8i)12-s + (−2.03e10 − 3.53e10i)13-s + (−3.12e11 + 1.80e11i)14-s + (−2.41e10 − 9.24e11i)15-s + (−1.37e11 + 2.38e11i)16-s − 1.69e12i·17-s + ⋯ |
| L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.477 − 0.878i)3-s + (0.249 + 0.433i)4-s + (1.38 − 0.801i)5-s + (−0.602 + 0.369i)6-s + (0.882 − 1.52i)7-s − 0.353i·8-s + (−0.544 − 0.838i)9-s − 1.13·10-s + (−0.167 − 0.0966i)11-s + (0.499 − 0.0130i)12-s + (−0.147 − 0.256i)13-s + (−1.08 + 0.624i)14-s + (−0.0419 − 1.60i)15-s + (−0.125 + 0.216i)16-s − 0.840i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.920 + 0.390i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (-0.920 + 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{21}{2})\) |
\(\approx\) |
\(2.739632598\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.739632598\) |
| \(L(11)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (627. + 362. i)T \) |
| 3 | \( 1 + (-2.81e4 + 5.18e4i)T \) |
| good | 5 | \( 1 + (-1.35e7 + 7.83e6i)T + (4.76e13 - 8.25e13i)T^{2} \) |
| 7 | \( 1 + (-2.49e8 + 4.31e8i)T + (-3.98e16 - 6.91e16i)T^{2} \) |
| 11 | \( 1 + (4.34e9 + 2.50e9i)T + (3.36e20 + 5.82e20i)T^{2} \) |
| 13 | \( 1 + (2.03e10 + 3.53e10i)T + (-9.50e21 + 1.64e22i)T^{2} \) |
| 17 | \( 1 + 1.69e12iT - 4.06e24T^{2} \) |
| 19 | \( 1 - 7.48e12T + 3.75e25T^{2} \) |
| 23 | \( 1 + (-1.47e13 + 8.50e12i)T + (8.58e26 - 1.48e27i)T^{2} \) |
| 29 | \( 1 + (-6.04e14 - 3.48e14i)T + (8.84e28 + 1.53e29i)T^{2} \) |
| 31 | \( 1 + (-7.10e14 - 1.23e15i)T + (-3.35e29 + 5.81e29i)T^{2} \) |
| 37 | \( 1 + 1.49e15T + 2.31e31T^{2} \) |
| 41 | \( 1 + (-9.85e15 + 5.69e15i)T + (9.00e31 - 1.56e32i)T^{2} \) |
| 43 | \( 1 + (1.43e16 - 2.47e16i)T + (-2.33e32 - 4.04e32i)T^{2} \) |
| 47 | \( 1 + (3.48e15 + 2.01e15i)T + (1.38e33 + 2.39e33i)T^{2} \) |
| 53 | \( 1 - 2.58e17iT - 3.05e34T^{2} \) |
| 59 | \( 1 + (1.54e17 - 8.94e16i)T + (1.30e35 - 2.26e35i)T^{2} \) |
| 61 | \( 1 + (2.85e17 - 4.95e17i)T + (-2.54e35 - 4.40e35i)T^{2} \) |
| 67 | \( 1 + (8.75e17 + 1.51e18i)T + (-1.66e36 + 2.87e36i)T^{2} \) |
| 71 | \( 1 - 2.21e18iT - 1.05e37T^{2} \) |
| 73 | \( 1 - 4.50e18T + 1.84e37T^{2} \) |
| 79 | \( 1 + (-2.31e18 + 4.00e18i)T + (-4.48e37 - 7.76e37i)T^{2} \) |
| 83 | \( 1 + (-1.46e19 - 8.47e18i)T + (1.20e38 + 2.08e38i)T^{2} \) |
| 89 | \( 1 - 4.89e19iT - 9.72e38T^{2} \) |
| 97 | \( 1 + (6.03e19 - 1.04e20i)T + (-2.71e39 - 4.70e39i)T^{2} \) |
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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
−13.61234923073069370058113534025, −12.22928990924014605325094663175, −10.55822380904297140802952738405, −9.319119631111292964887232722630, −8.052659095158716796248276293809, −6.86111736760333563334835013458, −4.97404778750487987289993540845, −2.83579528755553400803261786726, −1.24514725842094175080873563224, −1.01189088358025488581880923482,
1.88865253523086085893800555911, 2.70642456635042805980778986099, 5.12152947286707117951893697023, 6.10422072399631082061973259701, 8.125150953478441362835998500008, 9.308574361220479863283889833059, 10.16334130704058038315692609792, 11.51729506407849733617304828581, 13.81744864960704969716679430957, 14.78573466728838899026988237751
