L(s) = 1 | + (0.707 + 1.22i)2-s + (−0.499 + 0.866i)4-s + (−0.965 − 0.258i)5-s + i·7-s + (−0.366 − 1.36i)10-s + i·13-s + (−1.22 + 0.707i)14-s + (0.499 + 0.866i)16-s + (−0.707 + 1.22i)17-s + (0.707 − 0.707i)20-s + (0.866 + 0.499i)25-s + (−1.22 + 0.707i)26-s + (−0.866 − 0.499i)28-s − 1.41i·29-s + (0.5 − 0.866i)31-s + (−0.707 + 1.22i)32-s + ⋯ |
L(s) = 1 | + (0.707 + 1.22i)2-s + (−0.499 + 0.866i)4-s + (−0.965 − 0.258i)5-s + i·7-s + (−0.366 − 1.36i)10-s + i·13-s + (−1.22 + 0.707i)14-s + (0.499 + 0.866i)16-s + (−0.707 + 1.22i)17-s + (0.707 − 0.707i)20-s + (0.866 + 0.499i)25-s + (−1.22 + 0.707i)26-s + (−0.866 − 0.499i)28-s − 1.41i·29-s + (0.5 − 0.866i)31-s + (−0.707 + 1.22i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.660 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.660 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.233346779\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.233346779\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.965 + 0.258i)T \) |
| 7 | \( 1 - iT \) |
good | 2 | \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 - iT - T^{2} \) |
| 17 | \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + 1.41iT - T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + 1.41iT - T^{2} \) |
| 43 | \( 1 + iT - T^{2} \) |
| 47 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - 1.41iT - T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - 1.41T + T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + iT - T^{2} \) |
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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
−10.69483117177036742121311902740, −9.381289482233697668722366996649, −8.531685895417056732620443232688, −7.986695221952279478548030996542, −7.02839980733303439784427057113, −6.23620476636196969847436548663, −5.49480637324583059877835317335, −4.38773093338556766210026176799, −3.90472329811664455685918037025, −2.16347566830592833418594702396,
1.04732421159582330389029566494, 2.83767366652793367590392593449, 3.38481002041225447329156167167, 4.47622048953190884563003704876, 4.98680938332582365394871300663, 6.61763682452520045900206714826, 7.45480288850800164427673310506, 8.150843691133203713767395856717, 9.438785241409092119662161882844, 10.40992765833599898259609264822