L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + (−1.34 − 1.78i)5-s + (−0.499 − 0.866i)6-s + 4.03i·7-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (0.271 + 2.21i)10-s − 1.47·11-s + 0.999i·12-s + (−4.38 + 2.53i)13-s + (2.01 − 3.49i)14-s + (−0.271 − 2.21i)15-s + (−0.5 + 0.866i)16-s + (−0.0812 − 0.0469i)17-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.499 + 0.288i)3-s + (0.249 + 0.433i)4-s + (−0.601 − 0.798i)5-s + (−0.204 − 0.353i)6-s + 1.52i·7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (0.0859 + 0.701i)10-s − 0.443·11-s + 0.288i·12-s + (−1.21 + 0.701i)13-s + (0.539 − 0.933i)14-s + (−0.0701 − 0.573i)15-s + (−0.125 + 0.216i)16-s + (−0.0197 − 0.0113i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.435 - 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.435 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.314231 + 0.501097i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.314231 + 0.501097i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 3 | \( 1 + (-0.866 - 0.5i)T \) |
| 5 | \( 1 + (1.34 + 1.78i)T \) |
| 19 | \( 1 + (4.00 + 1.72i)T \) |
good | 7 | \( 1 - 4.03iT - 7T^{2} \) |
| 11 | \( 1 + 1.47T + 11T^{2} \) |
| 13 | \( 1 + (4.38 - 2.53i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (0.0812 + 0.0469i)T + (8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (2.22 - 1.28i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.10 - 3.64i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 4.26T + 31T^{2} \) |
| 37 | \( 1 - 1.53iT - 37T^{2} \) |
| 41 | \( 1 + (3.88 - 6.72i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.97 + 2.29i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.49 + 2.01i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (9.22 - 5.32i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.07 - 5.32i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.653 - 1.13i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-9.31 + 5.37i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.33 + 7.51i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-11.0 - 6.35i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (6.48 - 11.2i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 2.06iT - 83T^{2} \) |
| 89 | \( 1 + (0.813 + 1.40i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-10.3 - 5.97i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.03599505683573303945304194270, −9.839515630475239072838862774875, −9.223866774469694988515509660290, −8.496762761630007820686668815634, −7.927687694236427301598786046152, −6.68766454385059680316361987815, −5.24712476443241563713339547132, −4.43061960899959508433049649444, −2.95588211002608525063142582412, −1.98323027957190965900444698109,
0.36213723560067244779896110422, 2.33807003843672439927529322718, 3.59532228827846163443272033662, 4.70404714270378827474236292392, 6.34983060943065890119412852987, 7.10915105567034353799436291055, 7.80333076424960375743715994385, 8.277941097068598786986087482417, 9.848292221501339973541088900593, 10.27765467682152277182716580393