L(s) = 1 | + (5.65 − 9.79i)2-s + (40.5 − 23.3i)3-s + (−63.9 − 110. i)4-s + (−65.1 − 37.6i)5-s − 529. i·6-s + (2.24e3 − 843. i)7-s − 1.44e3·8-s + (1.09e3 − 1.89e3i)9-s + (−737. + 425. i)10-s + (−8.03e3 − 1.39e4i)11-s + (−5.18e3 − 2.99e3i)12-s − 1.63e4i·13-s + (4.44e3 − 2.67e4i)14-s − 3.52e3·15-s + (−8.19e3 + 1.41e4i)16-s + (−6.12e3 + 3.53e3i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (0.5 − 0.288i)3-s + (−0.249 − 0.433i)4-s + (−0.104 − 0.0602i)5-s − 0.408i·6-s + (0.936 − 0.351i)7-s − 0.353·8-s + (0.166 − 0.288i)9-s + (−0.0737 + 0.0425i)10-s + (−0.548 − 0.950i)11-s + (−0.249 − 0.144i)12-s − 0.572i·13-s + (0.115 − 0.697i)14-s − 0.0695·15-s + (−0.125 + 0.216i)16-s + (−0.0733 + 0.0423i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.727 + 0.686i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.727 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(0.873888 - 2.20067i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.873888 - 2.20067i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-5.65 + 9.79i)T \) |
| 3 | \( 1 + (-40.5 + 23.3i)T \) |
| 7 | \( 1 + (-2.24e3 + 843. i)T \) |
good | 5 | \( 1 + (65.1 + 37.6i)T + (1.95e5 + 3.38e5i)T^{2} \) |
| 11 | \( 1 + (8.03e3 + 1.39e4i)T + (-1.07e8 + 1.85e8i)T^{2} \) |
| 13 | \( 1 + 1.63e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + (6.12e3 - 3.53e3i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (3.59e4 + 2.07e4i)T + (8.49e9 + 1.47e10i)T^{2} \) |
| 23 | \( 1 + (1.52e4 - 2.64e4i)T + (-3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 + 4.96e5T + 5.00e11T^{2} \) |
| 31 | \( 1 + (2.33e5 - 1.34e5i)T + (4.26e11 - 7.38e11i)T^{2} \) |
| 37 | \( 1 + (-7.46e5 + 1.29e6i)T + (-1.75e12 - 3.04e12i)T^{2} \) |
| 41 | \( 1 + 3.80e5iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 3.70e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + (-6.05e6 - 3.49e6i)T + (1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 + (-2.69e6 - 4.67e6i)T + (-3.11e13 + 5.39e13i)T^{2} \) |
| 59 | \( 1 + (-5.16e6 + 2.98e6i)T + (7.34e13 - 1.27e14i)T^{2} \) |
| 61 | \( 1 + (-2.18e7 - 1.26e7i)T + (9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-1.57e7 - 2.73e7i)T + (-2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 + 2.18e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + (-2.52e7 + 1.45e7i)T + (4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (-1.26e7 + 2.19e7i)T + (-7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 + 2.26e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (1.00e8 + 5.79e7i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 + 3.41e7iT - 7.83e15T^{2} \) |
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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
−13.76391128911765091301673417646, −12.75725416679579006150824386486, −11.40473355511212214340996658339, −10.42372498254558904277650989692, −8.755886224305763245624070821718, −7.64321459399796406444116096081, −5.64466150086337152483465208269, −4.02386703040634880841378071536, −2.42747229743590644103152421069, −0.77236595191753250409894995144,
2.14748705475370895761836519408, 4.11299602629671579622778240645, 5.34371069149206671315904727901, 7.21401884384241471738971683744, 8.292082052158277893666429080157, 9.584171798389084390083798504417, 11.20415316722355259160132932503, 12.55272975145228598022939420714, 13.82395793014186460767590700940, 14.87601515668325943749531482708