L(s) = 1 | + 5.37i·2-s + (−2.08 + 4.75i)3-s − 20.8·4-s + 2.78·5-s + (−25.5 − 11.2i)6-s − 69.3i·8-s + (−18.3 − 19.8i)9-s + 14.9i·10-s − 17.5i·11-s + (43.5 − 99.4i)12-s + 47.8i·13-s + (−5.81 + 13.2i)15-s + 205.·16-s − 89.4·17-s + (106. − 98.3i)18-s + 42.2i·19-s + ⋯ |
L(s) = 1 | + 1.90i·2-s + (−0.401 + 0.915i)3-s − 2.61·4-s + 0.249·5-s + (−1.74 − 0.762i)6-s − 3.06i·8-s + (−0.677 − 0.735i)9-s + 0.473i·10-s − 0.481i·11-s + (1.04 − 2.39i)12-s + 1.02i·13-s + (−0.100 + 0.228i)15-s + 3.21·16-s − 1.27·17-s + (1.39 − 1.28i)18-s + 0.510i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 + 0.539i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.841 + 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.280638 - 0.0822421i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.280638 - 0.0822421i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (2.08 - 4.75i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 5.37iT - 8T^{2} \) |
| 5 | \( 1 - 2.78T + 125T^{2} \) |
| 11 | \( 1 + 17.5iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 47.8iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 89.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 42.2iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 87.6iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 40.8iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 95.6iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 64.5T + 5.06e4T^{2} \) |
| 41 | \( 1 - 403.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 230.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 365.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 598. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 236.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 430. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 428.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 519. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 764. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 227.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.13e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.13e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.48e3iT - 9.12e5T^{2} \) |
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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
−14.04657631613488927995091926184, −12.90127031284060428619919588775, −11.38583218848288886915512233968, −10.01620245073050989584600972854, −9.097865800975956824149706916075, −8.345852615919036906499218746160, −6.79482855669934488055806901825, −6.06362768241413878234655862410, −4.92710910199368791345363719965, −3.97049069948735871500573141960,
0.15110160268305911527600071334, 1.68469792340236696267768938165, 2.82059221615190546445660519975, 4.52836093399042612744680753427, 5.82007362294415497878478291494, 7.63148021232665564677512363088, 8.855328310629823930161313334570, 9.938578884932182106890663232748, 10.95871611261229522338655549458, 11.62075359998600227444631017672