| L(s) = 1 | + (4.89 − 2.82i)2-s + (25.6 − 8.43i)3-s + (15.9 − 27.7i)4-s + (1.59 + 0.922i)5-s + (101. − 113. i)6-s + (6.34 + 10.9i)7-s − 181. i·8-s + (586. − 432. i)9-s + 10.4·10-s + (−49.8 + 28.7i)11-s + (176. − 845. i)12-s + (−1.41e3 + 2.44e3i)13-s + (62.1 + 35.8i)14-s + (48.7 + 10.1i)15-s + (−512. − 886. i)16-s + 7.22e3i·17-s + ⋯ |
| L(s) = 1 | + (0.612 − 0.353i)2-s + (0.949 − 0.312i)3-s + (0.249 − 0.433i)4-s + (0.0127 + 0.00737i)5-s + (0.471 − 0.527i)6-s + (0.0184 + 0.0320i)7-s − 0.353i·8-s + (0.804 − 0.593i)9-s + 0.0104·10-s + (−0.0374 + 0.0216i)11-s + (0.102 − 0.489i)12-s + (−0.642 + 1.11i)13-s + (0.0226 + 0.0130i)14-s + (0.0144 + 0.00301i)15-s + (−0.125 − 0.216i)16-s + 1.46i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.724 + 0.689i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.724 + 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.36098 - 0.944083i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.36098 - 0.944083i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-4.89 + 2.82i)T \) |
| 3 | \( 1 + (-25.6 + 8.43i)T \) |
| good | 5 | \( 1 + (-1.59 - 0.922i)T + (7.81e3 + 1.35e4i)T^{2} \) |
| 7 | \( 1 + (-6.34 - 10.9i)T + (-5.88e4 + 1.01e5i)T^{2} \) |
| 11 | \( 1 + (49.8 - 28.7i)T + (8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 + (1.41e3 - 2.44e3i)T + (-2.41e6 - 4.18e6i)T^{2} \) |
| 17 | \( 1 - 7.22e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 1.06e4T + 4.70e7T^{2} \) |
| 23 | \( 1 + (-1.15e4 - 6.68e3i)T + (7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 + (-3.07e4 + 1.77e4i)T + (2.97e8 - 5.15e8i)T^{2} \) |
| 31 | \( 1 + (-8.65e3 + 1.49e4i)T + (-4.43e8 - 7.68e8i)T^{2} \) |
| 37 | \( 1 + 6.04e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + (7.54e4 + 4.35e4i)T + (2.37e9 + 4.11e9i)T^{2} \) |
| 43 | \( 1 + (-3.79e4 - 6.57e4i)T + (-3.16e9 + 5.47e9i)T^{2} \) |
| 47 | \( 1 + (-6.18e4 + 3.57e4i)T + (5.38e9 - 9.33e9i)T^{2} \) |
| 53 | \( 1 + 1.47e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + (-1.00e5 - 5.78e4i)T + (2.10e10 + 3.65e10i)T^{2} \) |
| 61 | \( 1 + (-2.01e4 - 3.48e4i)T + (-2.57e10 + 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-9.08e4 + 1.57e5i)T + (-4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 - 3.01e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 6.12e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + (-1.71e5 - 2.97e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + (1.31e5 - 7.59e4i)T + (1.63e11 - 2.83e11i)T^{2} \) |
| 89 | \( 1 - 1.02e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (-2.47e5 - 4.29e5i)T + (-4.16e11 + 7.21e11i)T^{2} \) |
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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
−17.24658019521784782259658905517, −15.39803937252827189608268853609, −14.43893579050471927713847743808, −13.25550783972698840405812192850, −12.07530547442694177273090832266, −10.20234104803373087620740184412, −8.547007156034662046228693818677, −6.65161925255404150004090982820, −4.13649598353590919107784415860, −2.10492963411147185629111956060,
2.89893191550281573396473599301, 4.86317093346278517272727683692, 7.21167299158374884369317297291, 8.729480083695416397887817718889, 10.47225791968756840191232791848, 12.51116138468688426295457208307, 13.72753207912969945392100949253, 14.88453984207508013941199817084, 15.80532557520805129867838457093, 17.29392126391587615726051532915
