| L(s) = 1 | + (−7.27 + 8.66i)2-s + (−71.9 − 37.1i)3-s + (−22.2 − 126. i)4-s + (242. − 667. i)5-s + (845. − 353. i)6-s + (702. − 3.98e3i)7-s + (1.25e3 + 724. i)8-s + (3.80e3 + 5.34e3i)9-s + (4.01e3 + 6.95e3i)10-s + (−675. − 1.85e3i)11-s + (−3.08e3 + 9.89e3i)12-s + (5.27e3 − 4.42e3i)13-s + (2.94e4 + 3.50e4i)14-s + (−4.22e4 + 3.89e4i)15-s + (−1.53e4 + 5.60e3i)16-s + (8.59e4 − 4.96e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.454 + 0.541i)2-s + (−0.888 − 0.458i)3-s + (−0.0868 − 0.492i)4-s + (0.388 − 1.06i)5-s + (0.652 − 0.272i)6-s + (0.292 − 1.65i)7-s + (0.306 + 0.176i)8-s + (0.579 + 0.815i)9-s + (0.401 + 0.695i)10-s + (−0.0461 − 0.126i)11-s + (−0.148 + 0.477i)12-s + (0.184 − 0.154i)13-s + (0.765 + 0.912i)14-s + (−0.834 + 0.770i)15-s + (−0.234 + 0.0855i)16-s + (1.02 − 0.594i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.821 + 0.570i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.821 + 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.297921 - 0.951113i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.297921 - 0.951113i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (7.27 - 8.66i)T \) |
| 3 | \( 1 + (71.9 + 37.1i)T \) |
| good | 5 | \( 1 + (-242. + 667. i)T + (-2.99e5 - 2.51e5i)T^{2} \) |
| 7 | \( 1 + (-702. + 3.98e3i)T + (-5.41e6 - 1.97e6i)T^{2} \) |
| 11 | \( 1 + (675. + 1.85e3i)T + (-1.64e8 + 1.37e8i)T^{2} \) |
| 13 | \( 1 + (-5.27e3 + 4.42e3i)T + (1.41e8 - 8.03e8i)T^{2} \) |
| 17 | \( 1 + (-8.59e4 + 4.96e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-7.98e4 + 1.38e5i)T + (-8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (3.83e5 - 6.76e4i)T + (7.35e10 - 2.67e10i)T^{2} \) |
| 29 | \( 1 + (-5.84e5 + 6.96e5i)T + (-8.68e10 - 4.92e11i)T^{2} \) |
| 31 | \( 1 + (-7.09e4 - 4.02e5i)T + (-8.01e11 + 2.91e11i)T^{2} \) |
| 37 | \( 1 + (1.78e6 + 3.09e6i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 + (-1.55e6 - 1.85e6i)T + (-1.38e12 + 7.86e12i)T^{2} \) |
| 43 | \( 1 + (5.76e6 - 2.09e6i)T + (8.95e12 - 7.51e12i)T^{2} \) |
| 47 | \( 1 + (-3.95e6 - 6.98e5i)T + (2.23e13 + 8.14e12i)T^{2} \) |
| 53 | \( 1 - 9.23e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + (-2.34e6 + 6.44e6i)T + (-1.12e14 - 9.43e13i)T^{2} \) |
| 61 | \( 1 + (4.42e6 - 2.50e7i)T + (-1.80e14 - 6.55e13i)T^{2} \) |
| 67 | \( 1 + (3.41e6 - 2.86e6i)T + (7.05e13 - 3.99e14i)T^{2} \) |
| 71 | \( 1 + (-1.78e7 + 1.03e7i)T + (3.22e14 - 5.59e14i)T^{2} \) |
| 73 | \( 1 + (4.93e6 - 8.54e6i)T + (-4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (2.85e6 + 2.39e6i)T + (2.63e14 + 1.49e15i)T^{2} \) |
| 83 | \( 1 + (-2.82e6 + 3.37e6i)T + (-3.91e14 - 2.21e15i)T^{2} \) |
| 89 | \( 1 + (-5.18e7 - 2.99e7i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 + (2.82e7 - 1.02e7i)T + (6.00e15 - 5.03e15i)T^{2} \) |
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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
−13.39459758177659909492720428588, −12.00587494182378387114147959203, −10.69840065538176900651555614794, −9.691455155439598835559730081909, −8.014092050433559868302566757968, −7.08294760078383474501326729969, −5.61448275406299435504806286656, −4.50352713414111150557129975963, −1.25819894759700699711819639935, −0.52223759005282823575074389053,
1.79372094690095291471864657033, 3.35374245251395298404125668504, 5.38687316894959664806331983044, 6.45628330455895063478363050743, 8.315352347971550299063801108107, 9.798862923455957304476295061814, 10.49634660511037665298233785108, 11.83677219371768758241477610190, 12.27873265765880220907455531988, 14.25995087164078488043399320204
