| L(s) = 1 | + (−3.16 − 1.82i)2-s + (7.01 + 13.9i)3-s + (−9.32 − 16.1i)4-s + (−37.1 + 64.3i)5-s + (3.22 − 56.8i)6-s + (45.3 + 121. i)7-s + 185. i·8-s + (−144. + 195. i)9-s + (235. − 135. i)10-s + (−34.1 + 19.6i)11-s + (159. − 243. i)12-s − 589. i·13-s + (78.5 − 467. i)14-s + (−1.15e3 − 65.5i)15-s + (40.0 − 69.3i)16-s + (−618. − 1.07e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.559 − 0.323i)2-s + (0.450 + 0.892i)3-s + (−0.291 − 0.504i)4-s + (−0.664 + 1.15i)5-s + (0.0365 − 0.645i)6-s + (0.349 + 0.936i)7-s + 1.02i·8-s + (−0.594 + 0.803i)9-s + (0.743 − 0.429i)10-s + (−0.0850 + 0.0490i)11-s + (0.319 − 0.487i)12-s − 0.967i·13-s + (0.107 − 0.637i)14-s + (−1.32 − 0.0752i)15-s + (0.0390 − 0.0676i)16-s + (−0.519 − 0.899i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.172 - 0.984i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.172 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.563812 + 0.671290i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.563812 + 0.671290i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-7.01 - 13.9i)T \) |
| 7 | \( 1 + (-45.3 - 121. i)T \) |
| good | 2 | \( 1 + (3.16 + 1.82i)T + (16 + 27.7i)T^{2} \) |
| 5 | \( 1 + (37.1 - 64.3i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 11 | \( 1 + (34.1 - 19.6i)T + (8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + 589. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + (618. + 1.07e3i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-2.39e3 - 1.38e3i)T + (1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-401. - 231. i)T + (3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 - 5.29e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + (-2.46e3 + 1.42e3i)T + (1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (3.92e3 - 6.79e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 - 1.21e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 5.35e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (3.37e3 - 5.85e3i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-1.38e4 + 8.00e3i)T + (2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + (3.87e3 + 6.70e3i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-4.86e3 - 2.80e3i)T + (4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-6.44e3 - 1.11e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + 6.06e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (-4.47e4 + 2.58e4i)T + (1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (3.36e4 - 5.83e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + 8.71e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + (-9.05e3 + 1.56e4i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 - 1.86e4iT - 8.58e9T^{2} \) |
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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
−17.93198994097899276598509955454, −15.84585616205252175652192760227, −14.97840719844376454325305291626, −14.09794583992014288153980342145, −11.57621696556357881973795598061, −10.57997457995044075839869512942, −9.370409651705026765462339849773, −7.946220753450467633229199490734, −5.29325543339507128221759007213, −2.92675739258410967540949824426,
0.77746130732839502851103233244, 4.15183998649131528845562652828, 7.14132311019952628939546318112, 8.167015087450145997991579594248, 9.206190495351708990084945284222, 11.70541506344921607214257448646, 12.91252586201716477904453469002, 13.87872967392343528488026321618, 15.84488028699232314842351114280, 16.98799321951188432791806418064
