L(s) = 1 | + (−0.5 + 0.866i)2-s + (4.5 + 7.79i)3-s + (15.5 + 26.8i)4-s + (17 − 29.4i)5-s − 9·6-s − 63·8-s + (−40.5 + 70.1i)9-s + (16.9 + 29.4i)10-s + (170 + 294. i)11-s + (−139.5 + 241. i)12-s + 454·13-s + 306·15-s + (−464.5 + 804. i)16-s + (399 + 691. i)17-s + (−40.5 − 70.1i)18-s + (−446 + 772. i)19-s + ⋯ |
L(s) = 1 | + (−0.0883 + 0.153i)2-s + (0.288 + 0.499i)3-s + (0.484 + 0.838i)4-s + (0.304 − 0.526i)5-s − 0.102·6-s − 0.348·8-s + (−0.166 + 0.288i)9-s + (0.0537 + 0.0931i)10-s + (0.423 + 0.733i)11-s + (−0.279 + 0.484i)12-s + 0.745·13-s + 0.351·15-s + (−0.453 + 0.785i)16-s + (0.334 + 0.579i)17-s + (−0.0294 − 0.0510i)18-s + (−0.283 + 0.490i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.266184886\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.266184886\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-4.5 - 7.79i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.5 - 0.866i)T + (-16 - 27.7i)T^{2} \) |
| 5 | \( 1 + (-17 + 29.4i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 11 | \( 1 + (-170 - 294. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 - 454T + 3.71e5T^{2} \) |
| 17 | \( 1 + (-399 - 691. i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (446 - 772. i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-1.59e3 + 2.76e3i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + 8.24e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + (-1.24e3 - 2.16e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + (4.89e3 - 8.48e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 - 1.98e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.72e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (4.46e3 - 7.73e3i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (75 + 129. i)T + (-2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 + (-2.11e4 - 3.67e4i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (7.37e3 - 1.27e4i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-838 - 1.45e3i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 - 1.45e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + (3.91e4 + 6.78e4i)T + (-1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (-1.13e3 + 1.96e3i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + 3.77e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (-5.86e4 + 1.01e5i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 - 1.00e4T + 8.58e9T^{2} \) |
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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
−12.58533218067522064031375841121, −11.46494446111996955188613008636, −10.40239734879337093154202076735, −9.145710049258621568770070955734, −8.429342519160791736982855892701, −7.25496542246421484664806699073, −5.99840586250065689225929036539, −4.45789763170981101844147039679, −3.30980044719199808460636837565, −1.71804856053153784102608933785,
0.76667496425119150520312119182, 2.08247390718390313233196119436, 3.40301642306557464678778925580, 5.49693741508349489769407815924, 6.41132485697829752175078152583, 7.37102860987291776611890731681, 8.859782024463065321493213918077, 9.759684604346111563874226656577, 11.04683385692800014174231081683, 11.46185058634540938349271985442