L(s) = 1 | + (0.707 + 0.707i)2-s + (−0.854 − 2.87i)3-s − 3i·4-s + (−4.57 + 2.01i)5-s + (1.42 − 2.63i)6-s + (−5.77 − 3.94i)7-s + (4.94 − 4.94i)8-s + (−7.53 + 4.91i)9-s + (−4.65 − 1.81i)10-s − 2.58i·11-s + (−8.62 + 2.56i)12-s + (8.94 − 8.94i)13-s + (−1.29 − 6.87i)14-s + (9.69 + 11.4i)15-s − 4.99·16-s + (0.581 − 0.581i)17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.353i)2-s + (−0.284 − 0.958i)3-s − 0.750i·4-s + (−0.915 + 0.402i)5-s + (0.238 − 0.439i)6-s + (−0.825 − 0.564i)7-s + (0.618 − 0.618i)8-s + (−0.837 + 0.546i)9-s + (−0.465 − 0.181i)10-s − 0.235i·11-s + (−0.718 + 0.213i)12-s + (0.687 − 0.687i)13-s + (−0.0924 − 0.491i)14-s + (0.646 + 0.763i)15-s − 0.312·16-s + (0.0342 − 0.0342i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.435 + 0.900i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.435 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.537658 - 0.857050i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.537658 - 0.857050i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.854 + 2.87i)T \) |
| 5 | \( 1 + (4.57 - 2.01i)T \) |
| 7 | \( 1 + (5.77 + 3.94i)T \) |
good | 2 | \( 1 + (-0.707 - 0.707i)T + 4iT^{2} \) |
| 11 | \( 1 + 2.58iT - 121T^{2} \) |
| 13 | \( 1 + (-8.94 + 8.94i)T - 169iT^{2} \) |
| 17 | \( 1 + (-0.581 + 0.581i)T - 289iT^{2} \) |
| 19 | \( 1 - 16.5T + 361T^{2} \) |
| 23 | \( 1 + (-26.5 + 26.5i)T - 529iT^{2} \) |
| 29 | \( 1 + 11.5T + 841T^{2} \) |
| 31 | \( 1 - 30.8iT - 961T^{2} \) |
| 37 | \( 1 + (41.4 - 41.4i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 17.1T + 1.68e3T^{2} \) |
| 43 | \( 1 + (25.1 + 25.1i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-45.2 + 45.2i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-34.1 + 34.1i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 - 47.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 78.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 + (-72.4 + 72.4i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 49.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-26.0 + 26.0i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 - 75.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (53.6 + 53.6i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 - 14.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (25.9 + 25.9i)T + 9.40e3iT^{2} \) |
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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
−13.33348179570851606855134764752, −12.31596076481823657240859206380, −11.05934597537128289841006317579, −10.31215025938571045663511953817, −8.529091656670237450245783684796, −7.14535410397616679544791953092, −6.58087011239751239142650295218, −5.21686618409417179869951748797, −3.35124504733449531519435395325, −0.72357733200806825756980922867,
3.22023649089247347666792604690, 4.10116427939317470885058391971, 5.44134118258553907755753535623, 7.22817697259111283402348790037, 8.674493517375120974848097251184, 9.452459602407238599129613192151, 11.09016717087511982605890558224, 11.72126453253888259989720832731, 12.58522358288910578437917910202, 13.65719600113482453404955081913