| 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
−16.52151643596982449819060843551, −14.93069298698949152009601886913, −13.69857061023250238155974070242, −11.45883575816539657922246455756, −10.44788659174046816721902677517, −9.900455364637909394774563480698, −7.43387934142993105974769481382, −5.40384860626584527576238071096, −3.64256599053856168247744302837, −1.82712971153826422243381594912,
0.69744578273320713254126785886, 1.55908432537436330140082286752, 5.25321244747136850969802242801, 6.02485914811783222229542139111, 8.047924810048138292992392903500, 9.074165035370277054166144101895, 11.30223908663603337509203924551, 12.87642770803337108581423967780, 13.68437880255331435964807376065, 15.79016272370505727930328591398
