| L(s) = 1 | + (−4 + 6.92i)2-s + (−10.4 − 18.0i)3-s + (−31.9 − 55.4i)4-s + (108. − 187. i)5-s + 167.·6-s + (726. − 544. i)7-s + 511.·8-s + (875. − 1.51e3i)9-s + (867. + 1.50e3i)10-s + (−1.29e3 − 2.23e3i)11-s + (−668. + 1.15e3i)12-s − 8.92e3·13-s + (864. + 7.20e3i)14-s − 4.52e3·15-s + (−2.04e3 + 3.54e3i)16-s + (5.55e3 + 9.62e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.223 − 0.386i)3-s + (−0.249 − 0.433i)4-s + (0.387 − 0.671i)5-s + 0.315·6-s + (0.800 − 0.599i)7-s + 0.353·8-s + (0.400 − 0.693i)9-s + (0.274 + 0.474i)10-s + (−0.292 − 0.506i)11-s + (−0.111 + 0.193i)12-s − 1.12·13-s + (0.0842 + 0.702i)14-s − 0.346·15-s + (−0.125 + 0.216i)16-s + (0.274 + 0.475i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.649 + 0.760i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.649 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.08525 - 0.500648i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.08525 - 0.500648i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (4 - 6.92i)T \) |
| 7 | \( 1 + (-726. + 544. i)T \) |
| good | 3 | \( 1 + (10.4 + 18.0i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + (-108. + 187. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 11 | \( 1 + (1.29e3 + 2.23e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + 8.92e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + (-5.55e3 - 9.62e3i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-4.32e3 + 7.48e3i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-3.30e4 + 5.72e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 - 1.28e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + (-1.02e5 - 1.76e5i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + (2.45e5 - 4.25e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + 6.23e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 4.22e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-6.02e5 + 1.04e6i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 + (-6.36e5 - 1.10e6i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + (8.40e5 + 1.45e6i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (1.06e6 - 1.83e6i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-1.67e6 - 2.90e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 - 2.49e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + (-1.63e6 - 2.83e6i)T + (-5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (-2.09e6 + 3.63e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 - 3.35e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (1.19e6 - 2.07e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 - 4.51e6T + 8.07e13T^{2} \) |
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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
−17.51853839580638365640454734518, −16.83967056340335110747271330353, −15.14173717594903611835961423072, −13.74202438414967161024787209666, −12.24784600466749305234349563932, −10.26846222914876209033288555047, −8.558999980883254884409565695148, −6.93589192782959242236176041601, −4.99384398925136809305952841458, −1.00219895995496005006123827351,
2.29469491037461225374195697173, 4.98274849897221237012060659966, 7.61522673375439852297053696182, 9.661066970369448145780138292809, 10.80553633908736571166543719922, 12.20394890856757011670640074478, 14.02446804471159408178211084410, 15.45871442304793656608144725321, 17.16793506059234785446559802056, 18.21807396171267329584407240681
