L(s) = 1 | + (0.927 − 1.06i)2-s + (−0.280 − 1.98i)4-s − 2i·5-s − 0.936·7-s + (−2.37 − 1.53i)8-s + (−2.13 − 1.85i)10-s + (−3.09 + 1.19i)11-s − 4.27·13-s + (−0.868 + 0.999i)14-s + (−3.84 + 1.11i)16-s − 3.33i·17-s + 2.89i·19-s + (−3.96 + 0.561i)20-s + (−1.58 + 4.41i)22-s − 5.12i·23-s + ⋯ |
L(s) = 1 | + (0.655 − 0.755i)2-s + (−0.140 − 0.990i)4-s − 0.894i·5-s − 0.353·7-s + (−0.839 − 0.543i)8-s + (−0.675 − 0.586i)10-s + (−0.932 + 0.361i)11-s − 1.18·13-s + (−0.232 + 0.267i)14-s + (−0.960 + 0.277i)16-s − 0.808i·17-s + 0.664i·19-s + (−0.885 + 0.125i)20-s + (−0.338 + 0.941i)22-s − 1.06i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 - 0.202i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.979 - 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.125775 + 1.22812i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.125775 + 1.22812i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.927 + 1.06i)T \) |
| 3 | \( 1 \) |
| 11 | \( 1 + (3.09 - 1.19i)T \) |
good | 5 | \( 1 + 2iT - 5T^{2} \) |
| 7 | \( 1 + 0.936T + 7T^{2} \) |
| 13 | \( 1 + 4.27T + 13T^{2} \) |
| 17 | \( 1 + 3.33iT - 17T^{2} \) |
| 19 | \( 1 - 2.89iT - 19T^{2} \) |
| 23 | \( 1 + 5.12iT - 23T^{2} \) |
| 29 | \( 1 - 6.60T + 29T^{2} \) |
| 31 | \( 1 + 1.73iT - 31T^{2} \) |
| 37 | \( 1 + 6.18iT - 37T^{2} \) |
| 41 | \( 1 + 1.46iT - 41T^{2} \) |
| 43 | \( 1 - 10.3iT - 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 + 8.24iT - 53T^{2} \) |
| 59 | \( 1 - 9.65T + 59T^{2} \) |
| 61 | \( 1 + 9.06T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 + 4.24iT - 71T^{2} \) |
| 73 | \( 1 + 13.2iT - 73T^{2} \) |
| 79 | \( 1 - 3.86T + 79T^{2} \) |
| 83 | \( 1 + 2.39iT - 83T^{2} \) |
| 89 | \( 1 - 3.47T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
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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
−9.960579314116078127606995395898, −9.264713272982742262428283522475, −8.251047416618736331816970458393, −7.14378820159148612830620008794, −6.03564141384180499154768906295, −4.91421147615412089770067671177, −4.63384944886600251641453491881, −3.11593781446556235103838107406, −2.14778982151618104855725758921, −0.46550300119437706498428161548,
2.60585713885400929945633414482, 3.26123572230603238888907430760, 4.58357982854319480849363911288, 5.47459274427246635757809087977, 6.44162738053346898921590320365, 7.15703193874331400598437051624, 7.88724694585704521258103127611, 8.837408350390453684665692586056, 9.956320010210280616128099944928, 10.70959221308989809050965042764