L(s) = 1 | + (−64 + 110. i)2-s + (−2.28e3 − 3.94e3i)3-s + (−8.19e3 − 1.41e4i)4-s + (1.51e5 − 2.63e5i)5-s + 5.83e5·6-s + (1.80e6 + 1.22e6i)7-s + 2.09e6·8-s + (−3.22e6 + 5.58e6i)9-s + (1.94e7 + 3.36e7i)10-s + (−4.22e7 − 7.31e7i)11-s + (−3.73e7 + 6.47e7i)12-s + 2.50e8·13-s + (−2.51e8 + 1.21e8i)14-s − 1.38e9·15-s + (−1.34e8 + 2.32e8i)16-s + (7.61e8 + 1.31e9i)17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.602 − 1.04i)3-s + (−0.249 − 0.433i)4-s + (0.869 − 1.50i)5-s + 0.851·6-s + (0.826 + 0.563i)7-s + 0.353·8-s + (−0.224 + 0.389i)9-s + (0.614 + 1.06i)10-s + (−0.653 − 1.13i)11-s + (−0.301 + 0.521i)12-s + 1.10·13-s + (−0.637 + 0.306i)14-s − 2.09·15-s + (−0.125 + 0.216i)16-s + (0.450 + 0.780i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.509 + 0.860i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (-0.509 + 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(8)\) |
\(\approx\) |
\(0.725528 - 1.27333i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.725528 - 1.27333i\) |
\(L(\frac{17}{2})\) |
|
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 | \( 1 + (-1.80e6 - 1.22e6i)T \) |
good | 3 | \( 1 + (2.28e3 + 3.94e3i)T + (-7.17e6 + 1.24e7i)T^{2} \) |
| 5 | \( 1 + (-1.51e5 + 2.63e5i)T + (-1.52e10 - 2.64e10i)T^{2} \) |
| 11 | \( 1 + (4.22e7 + 7.31e7i)T + (-2.08e15 + 3.61e15i)T^{2} \) |
| 13 | \( 1 - 2.50e8T + 5.11e16T^{2} \) |
| 17 | \( 1 + (-7.61e8 - 1.31e9i)T + (-1.43e18 + 2.47e18i)T^{2} \) |
| 19 | \( 1 + (-3.41e9 + 5.90e9i)T + (-7.59e18 - 1.31e19i)T^{2} \) |
| 23 | \( 1 + (2.60e8 - 4.51e8i)T + (-1.33e20 - 2.30e20i)T^{2} \) |
| 29 | \( 1 + 9.79e10T + 8.62e21T^{2} \) |
| 31 | \( 1 + (1.45e10 + 2.52e10i)T + (-1.17e22 + 2.03e22i)T^{2} \) |
| 37 | \( 1 + (1.03e11 - 1.78e11i)T + (-1.66e23 - 2.88e23i)T^{2} \) |
| 41 | \( 1 + 6.30e11T + 1.55e24T^{2} \) |
| 43 | \( 1 - 2.95e11T + 3.17e24T^{2} \) |
| 47 | \( 1 + (2.25e12 - 3.90e12i)T + (-6.03e24 - 1.04e25i)T^{2} \) |
| 53 | \( 1 + (-4.51e12 - 7.81e12i)T + (-3.65e25 + 6.33e25i)T^{2} \) |
| 59 | \( 1 + (8.01e12 + 1.38e13i)T + (-1.82e26 + 3.16e26i)T^{2} \) |
| 61 | \( 1 + (-1.72e13 + 2.98e13i)T + (-3.01e26 - 5.21e26i)T^{2} \) |
| 67 | \( 1 + (-5.56e12 - 9.63e12i)T + (-1.23e27 + 2.13e27i)T^{2} \) |
| 71 | \( 1 + 6.59e13T + 5.87e27T^{2} \) |
| 73 | \( 1 + (-6.58e13 - 1.14e14i)T + (-4.45e27 + 7.71e27i)T^{2} \) |
| 79 | \( 1 + (-1.10e14 + 1.91e14i)T + (-1.45e28 - 2.52e28i)T^{2} \) |
| 83 | \( 1 + 7.69e13T + 6.11e28T^{2} \) |
| 89 | \( 1 + (-2.20e13 + 3.82e13i)T + (-8.70e28 - 1.50e29i)T^{2} \) |
| 97 | \( 1 + 3.96e14T + 6.33e29T^{2} \) |
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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
−15.79016272370505727930328591398, −13.68437880255331435964807376065, −12.87642770803337108581423967780, −11.30223908663603337509203924551, −9.074165035370277054166144101895, −8.047924810048138292992392903500, −6.02485914811783222229542139111, −5.25321244747136850969802242801, −1.55908432537436330140082286752, −0.69744578273320713254126785886,
1.82712971153826422243381594912, 3.64256599053856168247744302837, 5.40384860626584527576238071096, 7.43387934142993105974769481382, 9.900455364637909394774563480698, 10.44788659174046816721902677517, 11.45883575816539657922246455756, 13.69857061023250238155974070242, 14.93069298698949152009601886913, 16.52151643596982449819060843551