L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.922 − 1.59i)3-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s − 1.84·6-s + (2.59 + 0.506i)7-s + 0.999·8-s + (−0.201 − 0.349i)9-s + (−0.499 + 0.866i)10-s + (−0.5 + 0.866i)11-s + (0.922 + 1.59i)12-s + 4.07·13-s + (−0.859 − 2.50i)14-s − 1.84·15-s + (−0.5 − 0.866i)16-s + (3.51 − 6.09i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.532 − 0.922i)3-s + (−0.249 + 0.433i)4-s + (−0.223 − 0.387i)5-s − 0.753·6-s + (0.981 + 0.191i)7-s + 0.353·8-s + (−0.0672 − 0.116i)9-s + (−0.158 + 0.273i)10-s + (−0.150 + 0.261i)11-s + (0.266 + 0.461i)12-s + 1.12·13-s + (−0.229 − 0.668i)14-s − 0.476·15-s + (−0.125 − 0.216i)16-s + (0.853 − 1.47i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.253 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.253 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.03197 - 1.33685i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.03197 - 1.33685i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (-2.59 - 0.506i)T \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
good | 3 | \( 1 + (-0.922 + 1.59i)T + (-1.5 - 2.59i)T^{2} \) |
| 13 | \( 1 - 4.07T + 13T^{2} \) |
| 17 | \( 1 + (-3.51 + 6.09i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.22 - 2.11i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.78 + 4.81i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 0.245T + 29T^{2} \) |
| 31 | \( 1 + (2.22 - 3.84i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.79 + 4.84i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 8.26T + 41T^{2} \) |
| 43 | \( 1 + 1.56T + 43T^{2} \) |
| 47 | \( 1 + (-2.44 - 4.22i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2.77 - 4.80i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.15 + 3.73i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.92 + 3.33i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.34 + 2.32i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 + (8.38 - 14.5i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.25 + 9.09i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 + (1.23 + 2.14i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 6.25T + 97T^{2} \) |
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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
−10.11427452286958924959815059081, −8.992519471073896349417565037904, −8.334129055301956100250769474471, −7.76081499787699674310022594681, −6.95224872129033583123913347619, −5.49241753367894110711763420995, −4.51462760603990255935035671783, −3.21429221262610674295723559414, −2.02037818854087896694741839382, −1.09529631868568785232808973671,
1.51720719381139141444660881168, 3.42213953609200855325507197110, 4.06544077304013137284178778275, 5.23908467954794309333585065399, 6.15600522351451622055389234147, 7.29613102277600221770842779115, 8.280607132329935425079001239402, 8.579908723886539961550378219609, 9.739136453909784640324751557067, 10.38518349800459894635739896976