| L(s) = 1 | + (1.36 + 0.366i)2-s + (0.866 + 1.5i)3-s + (1.73 + i)4-s + (4.02 − 4.02i)5-s + (0.633 + 2.36i)6-s + (−4.99 + 1.33i)7-s + (1.99 + 2i)8-s + (−1.5 + 2.59i)9-s + (6.97 − 4.02i)10-s + (−3.41 + 12.7i)11-s + 3.46i·12-s + (3.81 − 12.4i)13-s − 7.31·14-s + (9.52 + 2.55i)15-s + (1.99 + 3.46i)16-s + (−16.3 − 9.42i)17-s + ⋯ |
| L(s) = 1 | + (0.683 + 0.183i)2-s + (0.288 + 0.5i)3-s + (0.433 + 0.250i)4-s + (0.805 − 0.805i)5-s + (0.105 + 0.394i)6-s + (−0.714 + 0.191i)7-s + (0.249 + 0.250i)8-s + (−0.166 + 0.288i)9-s + (0.697 − 0.402i)10-s + (−0.310 + 1.15i)11-s + 0.288i·12-s + (0.293 − 0.956i)13-s − 0.522·14-s + (0.635 + 0.170i)15-s + (0.124 + 0.216i)16-s + (−0.960 − 0.554i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.445i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.895 - 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.93001 + 0.453849i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.93001 + 0.453849i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.36 - 0.366i)T \) |
| 3 | \( 1 + (-0.866 - 1.5i)T \) |
| 13 | \( 1 + (-3.81 + 12.4i)T \) |
| good | 5 | \( 1 + (-4.02 + 4.02i)T - 25iT^{2} \) |
| 7 | \( 1 + (4.99 - 1.33i)T + (42.4 - 24.5i)T^{2} \) |
| 11 | \( 1 + (3.41 - 12.7i)T + (-104. - 60.5i)T^{2} \) |
| 17 | \( 1 + (16.3 + 9.42i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (7.85 + 29.3i)T + (-312. + 180.5i)T^{2} \) |
| 23 | \( 1 + (10.4 - 6.05i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-18.4 - 31.9i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (6.48 - 6.48i)T - 961iT^{2} \) |
| 37 | \( 1 + (-13.6 + 50.7i)T + (-1.18e3 - 684.5i)T^{2} \) |
| 41 | \( 1 + (6.38 + 1.70i)T + (1.45e3 + 840.5i)T^{2} \) |
| 43 | \( 1 + (-11.7 - 6.80i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-61.6 - 61.6i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 - 4.64T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-24.1 + 6.45i)T + (3.01e3 - 1.74e3i)T^{2} \) |
| 61 | \( 1 + (19.2 - 33.3i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-91.8 - 24.6i)T + (3.88e3 + 2.24e3i)T^{2} \) |
| 71 | \( 1 + (-10.5 - 39.2i)T + (-4.36e3 + 2.52e3i)T^{2} \) |
| 73 | \( 1 + (-60.4 - 60.4i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 + 94.2T + 6.24e3T^{2} \) |
| 83 | \( 1 + (63.4 - 63.4i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + (-40.3 + 150. i)T + (-6.85e3 - 3.96e3i)T^{2} \) |
| 97 | \( 1 + (39.2 + 146. i)T + (-8.14e3 + 4.70e3i)T^{2} \) |
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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
−14.19669188061488014068221775101, −13.03239388779931940198648714943, −12.70188107472452708306478363051, −10.95081036181626650825447065815, −9.699519629941042810164147850321, −8.806611844648295471473901094277, −7.07945159541479796051819060587, −5.58224358065505962444410043928, −4.52559726024657033911490971382, −2.60663425455804175641434139017,
2.26404979982670582837484680998, 3.75848604698402290349650874034, 6.08742647797107526928395949237, 6.55011400135775077158026706076, 8.328557056663830287523161360047, 9.892924643009544628558941781215, 10.86763285361494331761247741347, 12.12049479701729260829120512037, 13.45267820216474587961072514927, 13.77970767469074200095028040316
