| L(s) = 1 | + (−7.46 − 2.87i)2-s + (−26.9 + 1.24i)3-s + (47.4 + 42.9i)4-s + (−7.54 + 13.0i)5-s + (204. + 68.3i)6-s + (−156. − 271. i)7-s + (−230. − 457. i)8-s + (725. − 67.1i)9-s + (93.9 − 75.8i)10-s + (−66.1 − 114. i)11-s + (−1.33e3 − 1.10e3i)12-s + (−3.14e3 − 1.81e3i)13-s + (387. + 2.47e3i)14-s + (187. − 361. i)15-s + (401. + 4.07e3i)16-s − 9.66e3i·17-s + ⋯ |
| L(s) = 1 | + (−0.932 − 0.359i)2-s + (−0.998 + 0.0460i)3-s + (0.740 + 0.671i)4-s + (−0.0603 + 0.104i)5-s + (0.948 + 0.316i)6-s + (−0.456 − 0.790i)7-s + (−0.449 − 0.893i)8-s + (0.995 − 0.0920i)9-s + (0.0939 − 0.0758i)10-s + (−0.0497 − 0.0860i)11-s + (−0.771 − 0.636i)12-s + (−1.43 − 0.827i)13-s + (0.141 + 0.901i)14-s + (0.0554 − 0.107i)15-s + (0.0980 + 0.995i)16-s − 1.96i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.308 - 0.951i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.308 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.258245 + 0.187630i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.258245 + 0.187630i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (7.46 + 2.87i)T \) |
| 3 | \( 1 + (26.9 - 1.24i)T \) |
| good | 5 | \( 1 + (7.54 - 13.0i)T + (-7.81e3 - 1.35e4i)T^{2} \) |
| 7 | \( 1 + (156. + 271. i)T + (-5.88e4 + 1.01e5i)T^{2} \) |
| 11 | \( 1 + (66.1 + 114. i)T + (-8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 + (3.14e3 + 1.81e3i)T + (2.41e6 + 4.18e6i)T^{2} \) |
| 17 | \( 1 + 9.66e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 8.07e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 + (-1.27e4 - 7.34e3i)T + (7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 + (-2.82e3 - 4.88e3i)T + (-2.97e8 + 5.15e8i)T^{2} \) |
| 31 | \( 1 + (1.01e4 - 1.75e4i)T + (-4.43e8 - 7.68e8i)T^{2} \) |
| 37 | \( 1 - 2.49e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + (-2.50e4 - 1.44e4i)T + (2.37e9 + 4.11e9i)T^{2} \) |
| 43 | \( 1 + (9.08e4 - 5.24e4i)T + (3.16e9 - 5.47e9i)T^{2} \) |
| 47 | \( 1 + (1.32e5 - 7.63e4i)T + (5.38e9 - 9.33e9i)T^{2} \) |
| 53 | \( 1 - 1.88e5T + 2.21e10T^{2} \) |
| 59 | \( 1 + (5.88e3 - 1.01e4i)T + (-2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (2.50e5 - 1.44e5i)T + (2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-2.87e5 - 1.65e5i)T + (4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 + 1.84e4iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 2.54e4T + 1.51e11T^{2} \) |
| 79 | \( 1 + (1.61e5 + 2.79e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + (-3.33e5 - 5.77e5i)T + (-1.63e11 + 2.83e11i)T^{2} \) |
| 89 | \( 1 - 3.94e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (-5.31e5 - 9.20e5i)T + (-4.16e11 + 7.21e11i)T^{2} \) |
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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
−13.21986385328811337366388159746, −12.22420498103545136792758857740, −11.27241631089452700919047233018, −10.21162205200166479872040835234, −9.531954282959579081683423238567, −7.55355336303970818225995351686, −6.89969892620437451665857014103, −5.10533519887544069142010838352, −3.15681668847495150428539301568, −0.975788643863672771699810460961,
0.23958066846989386660956475112, 2.12517235862846007520559132700, 4.87914851097986029720203112702, 6.20816479801085463658753107347, 7.10185036176754439820727267503, 8.681207887889812349579035657804, 9.790059135980326298796700060258, 10.82049628374048349106404842356, 11.94973934497348157577097109391, 12.80124264141088130535331827741
