| L(s) = 1 | + (5.62 + 11.6i)2-s + (−16.3 + 34.0i)3-s + (−65.0 + 81.5i)4-s + (15.2 + 3.48i)5-s − 490.·6-s + 124. i·7-s + (−510. − 116. i)8-s + (−434. − 545. i)9-s + (45.2 + 198. i)10-s + (726. + 911. i)11-s + (−1.71e3 − 3.55e3i)12-s + (539. − 2.36e3i)13-s + (−1.45e3 + 701. i)14-s + (−369. + 463. i)15-s + (−25.4 − 111. i)16-s + (−273. − 1.19e3i)17-s + ⋯ |
| L(s) = 1 | + (0.703 + 1.46i)2-s + (−0.606 + 1.26i)3-s + (−1.01 + 1.27i)4-s + (0.122 + 0.0279i)5-s − 2.26·6-s + 0.363i·7-s + (−0.996 − 0.227i)8-s + (−0.596 − 0.748i)9-s + (0.0452 + 0.198i)10-s + (0.545 + 0.684i)11-s + (−0.989 − 2.05i)12-s + (0.245 − 1.07i)13-s + (−0.531 + 0.255i)14-s + (−0.109 + 0.137i)15-s + (−0.00621 − 0.0272i)16-s + (−0.0556 − 0.243i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.563 + 0.826i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.563 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.841335 - 1.59239i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.841335 - 1.59239i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (5.62e3 - 7.93e4i)T \) |
| good | 2 | \( 1 + (-5.62 - 11.6i)T + (-39.9 + 50.0i)T^{2} \) |
| 3 | \( 1 + (16.3 - 34.0i)T + (-454. - 569. i)T^{2} \) |
| 5 | \( 1 + (-15.2 - 3.48i)T + (1.40e4 + 6.77e3i)T^{2} \) |
| 7 | \( 1 - 124. iT - 1.17e5T^{2} \) |
| 11 | \( 1 + (-726. - 911. i)T + (-3.94e5 + 1.72e6i)T^{2} \) |
| 13 | \( 1 + (-539. + 2.36e3i)T + (-4.34e6 - 2.09e6i)T^{2} \) |
| 17 | \( 1 + (273. + 1.19e3i)T + (-2.17e7 + 1.04e7i)T^{2} \) |
| 19 | \( 1 + (-3.35e3 - 2.67e3i)T + (1.04e7 + 4.58e7i)T^{2} \) |
| 23 | \( 1 + (-3.33e3 - 4.18e3i)T + (-3.29e7 + 1.44e8i)T^{2} \) |
| 29 | \( 1 + (4.81e3 + 1.00e4i)T + (-3.70e8 + 4.65e8i)T^{2} \) |
| 31 | \( 1 + (3.19e4 - 1.53e4i)T + (5.53e8 - 6.93e8i)T^{2} \) |
| 37 | \( 1 - 5.24e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + (-7.33e4 + 3.53e4i)T + (2.96e9 - 3.71e9i)T^{2} \) |
| 47 | \( 1 + (-8.67e4 + 1.08e5i)T + (-2.39e9 - 1.05e10i)T^{2} \) |
| 53 | \( 1 + (-2.76e4 - 1.21e5i)T + (-1.99e10 + 9.61e9i)T^{2} \) |
| 59 | \( 1 + (-5.52e4 - 2.41e5i)T + (-3.80e10 + 1.83e10i)T^{2} \) |
| 61 | \( 1 + (-4.77e3 + 9.91e3i)T + (-3.21e10 - 4.02e10i)T^{2} \) |
| 67 | \( 1 + (-1.88e5 + 2.36e5i)T + (-2.01e10 - 8.81e10i)T^{2} \) |
| 71 | \( 1 + (4.90e5 + 3.91e5i)T + (2.85e10 + 1.24e11i)T^{2} \) |
| 73 | \( 1 + (5.90e4 + 1.34e4i)T + (1.36e11 + 6.56e10i)T^{2} \) |
| 79 | \( 1 - 7.50e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + (-6.16e5 - 2.96e5i)T + (2.03e11 + 2.55e11i)T^{2} \) |
| 89 | \( 1 + (-2.32e4 + 4.81e4i)T + (-3.09e11 - 3.88e11i)T^{2} \) |
| 97 | \( 1 + (-1.41e5 - 1.76e5i)T + (-1.85e11 + 8.12e11i)T^{2} \) |
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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
−15.43358905816995755272976122011, −14.83302320881036403135164196232, −13.50793779994390395855627783249, −12.08242703650916918644099020204, −10.51944400003803468686173065198, −9.231425193513424747476181197754, −7.60728511409921899640668560836, −5.99398655506110415235445978546, −5.12110050976513703734310060399, −3.86866040536690978451855840896,
0.77178522870767497394159389346, 1.96287787441470213383798720642, 3.90765506143148151415175969690, 5.75846294076892547351480846258, 7.20311626125065307295598587475, 9.284373092824561888033110998636, 10.99300916253510614330987314683, 11.61263553047146701871955535626, 12.64572431883423808417677759858, 13.49897570546888442916085111361
