| L(s) = 1 | + (1.73 − 1.00i)2-s + (34.6 − 31.4i)3-s + (−61.9 + 107. i)4-s + (181. + 313. i)5-s + (28.5 − 89.1i)6-s + (799. − 429. i)7-s + 504. i·8-s + (208. − 2.17e3i)9-s + (628. + 362. i)10-s + (4.79e3 + 2.76e3i)11-s + (1.23e3 + 5.66e3i)12-s + 1.89e3i·13-s + (954. − 1.54e3i)14-s + (1.61e4 + 5.16e3i)15-s + (−7.43e3 − 1.28e4i)16-s + (4.42e3 − 7.67e3i)17-s + ⋯ |
| L(s) = 1 | + (0.153 − 0.0884i)2-s + (0.740 − 0.672i)3-s + (−0.484 + 0.838i)4-s + (0.648 + 1.12i)5-s + (0.0538 − 0.168i)6-s + (0.880 − 0.473i)7-s + 0.348i·8-s + (0.0955 − 0.995i)9-s + (0.198 + 0.114i)10-s + (1.08 + 0.627i)11-s + (0.205 + 0.946i)12-s + 0.239i·13-s + (0.0930 − 0.150i)14-s + (1.23 + 0.395i)15-s + (−0.453 − 0.785i)16-s + (0.218 − 0.378i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.305i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.952 - 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.28741 + 0.357978i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.28741 + 0.357978i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-34.6 + 31.4i)T \) |
| 7 | \( 1 + (-799. + 429. i)T \) |
| good | 2 | \( 1 + (-1.73 + 1.00i)T + (64 - 110. i)T^{2} \) |
| 5 | \( 1 + (-181. - 313. i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 11 | \( 1 + (-4.79e3 - 2.76e3i)T + (9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 - 1.89e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 + (-4.42e3 + 7.67e3i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (3.21e4 - 1.85e4i)T + (4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (8.32e3 - 4.80e3i)T + (1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + 7.72e4iT - 1.72e10T^{2} \) |
| 31 | \( 1 + (2.10e5 + 1.21e5i)T + (1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + (-6.65e4 - 1.15e5i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 - 7.04e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 4.60e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (3.77e4 + 6.53e4i)T + (-2.53e11 + 4.38e11i)T^{2} \) |
| 53 | \( 1 + (1.76e6 + 1.02e6i)T + (5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + (-1.23e6 + 2.13e6i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-1.09e6 + 6.34e5i)T + (1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-1.15e6 + 1.99e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 - 5.32e6iT - 9.09e12T^{2} \) |
| 73 | \( 1 + (2.78e6 + 1.61e6i)T + (5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (-2.74e6 - 4.74e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + 4.19e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (-9.21e5 - 1.59e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 - 3.07e6iT - 8.07e13T^{2} \) |
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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
−17.11535362682605844241155912692, −14.56302828391873264118124484177, −14.25087654199518449952235086092, −12.89699431381013646457066173049, −11.47424787051702363551143741442, −9.558008983553131783744016451665, −7.987658655189919221130350584618, −6.75142869870750698727955080798, −3.90618083121950635873690642848, −2.08912429366545354768969684978,
1.52867784632677600057185926116, 4.42560222249440108189528197549, 5.63259887280075204306745959964, 8.653952011469703455873751211444, 9.231576737989266030012910260334, 10.82704691225306826760130902408, 12.91511367557889183725310363573, 14.17346504948843780032628230195, 14.90717199682183573508940537304, 16.38534250650980298408356115377
