| L(s) = 1 | + (−45.2 + 78.3i)2-s + (−746. + 431. i)3-s + (−4.09e3 − 7.09e3i)4-s + (4.57e4 + 2.64e4i)5-s − 7.80e4i·6-s + (−6.02e5 + 5.61e5i)7-s + 7.41e5·8-s + (−2.01e6 + 3.49e6i)9-s + (−4.14e6 + 2.39e6i)10-s + (−1.49e7 − 2.59e7i)11-s + (6.11e6 + 3.53e6i)12-s + 6.10e7i·13-s + (−1.67e7 − 7.26e7i)14-s − 4.55e7·15-s + (−3.35e7 + 5.81e7i)16-s + (6.07e8 − 3.50e8i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.341 + 0.197i)3-s + (−0.249 − 0.433i)4-s + (0.585 + 0.338i)5-s − 0.278i·6-s + (−0.731 + 0.682i)7-s + 0.353·8-s + (−0.422 + 0.731i)9-s + (−0.414 + 0.239i)10-s + (−0.769 − 1.33i)11-s + (0.170 + 0.0985i)12-s + 0.973i·13-s + (−0.159 − 0.688i)14-s − 0.266·15-s + (−0.125 + 0.216i)16-s + (1.48 − 0.855i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.100 + 0.994i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{15}{2})\) |
\(\approx\) |
\(0.215578 - 0.194924i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.215578 - 0.194924i\) |
| \(L(8)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (45.2 - 78.3i)T \) |
| 7 | \( 1 + (6.02e5 - 5.61e5i)T \) |
| good | 3 | \( 1 + (746. - 431. i)T + (2.39e6 - 4.14e6i)T^{2} \) |
| 5 | \( 1 + (-4.57e4 - 2.64e4i)T + (3.05e9 + 5.28e9i)T^{2} \) |
| 11 | \( 1 + (1.49e7 + 2.59e7i)T + (-1.89e14 + 3.28e14i)T^{2} \) |
| 13 | \( 1 - 6.10e7iT - 3.93e15T^{2} \) |
| 17 | \( 1 + (-6.07e8 + 3.50e8i)T + (8.41e16 - 1.45e17i)T^{2} \) |
| 19 | \( 1 + (8.10e8 + 4.67e8i)T + (3.99e17 + 6.91e17i)T^{2} \) |
| 23 | \( 1 + (-2.67e9 + 4.62e9i)T + (-5.79e18 - 1.00e19i)T^{2} \) |
| 29 | \( 1 + 2.60e10T + 2.97e20T^{2} \) |
| 31 | \( 1 + (-1.46e10 + 8.44e9i)T + (3.78e20 - 6.55e20i)T^{2} \) |
| 37 | \( 1 + (1.69e8 - 2.93e8i)T + (-4.50e21 - 7.80e21i)T^{2} \) |
| 41 | \( 1 - 2.65e11iT - 3.79e22T^{2} \) |
| 43 | \( 1 + 1.33e11T + 7.38e22T^{2} \) |
| 47 | \( 1 + (3.25e11 + 1.87e11i)T + (1.28e23 + 2.22e23i)T^{2} \) |
| 53 | \( 1 + (7.07e11 + 1.22e12i)T + (-6.89e23 + 1.19e24i)T^{2} \) |
| 59 | \( 1 + (-1.85e12 + 1.07e12i)T + (3.09e24 - 5.36e24i)T^{2} \) |
| 61 | \( 1 + (1.76e12 + 1.02e12i)T + (4.93e24 + 8.55e24i)T^{2} \) |
| 67 | \( 1 + (-3.51e12 - 6.08e12i)T + (-1.83e25 + 3.18e25i)T^{2} \) |
| 71 | \( 1 - 1.32e12T + 8.27e25T^{2} \) |
| 73 | \( 1 + (1.14e13 - 6.58e12i)T + (6.10e25 - 1.05e26i)T^{2} \) |
| 79 | \( 1 + (2.22e12 - 3.84e12i)T + (-1.84e26 - 3.19e26i)T^{2} \) |
| 83 | \( 1 - 2.90e13iT - 7.36e26T^{2} \) |
| 89 | \( 1 + (9.69e11 + 5.59e11i)T + (9.78e26 + 1.69e27i)T^{2} \) |
| 97 | \( 1 + 3.91e13iT - 6.52e27T^{2} \) |
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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
−16.21166631893390172270193761691, −14.51268947216488739946887103262, −13.28296726106229072444992330105, −11.24689511074762611832575805799, −9.854277810033312695116677641693, −8.390646122019977055703186274193, −6.41652225710180331269964816395, −5.31032519890985023224464662578, −2.63911518138859897580802014608, −0.13145027009017257288098280500,
1.43209136641335523416168528939, 3.43614159799666821013766553518, 5.63087236334585483109138509674, 7.54754075056860722819510820294, 9.525880319486179390301857607995, 10.49108663149756387429590452876, 12.40561081378197525743486732435, 13.12193858928838964166501303142, 15.07365242161651476826428288892, 17.01134647349158626896569990066
