| L(s) = 1 | + (16 − 27.7i)2-s + (121.5 + 210. i)3-s + (−511. − 886. i)4-s + (−4.43e3 + 7.68e3i)5-s + 7.77e3·6-s + (−1.44e4 + 4.20e4i)7-s − 3.27e4·8-s + (−2.95e4 + 5.11e4i)9-s + (1.42e5 + 2.46e5i)10-s + (−3.34e5 − 5.79e5i)11-s + (1.24e5 − 2.15e5i)12-s + 8.13e5·13-s + (9.34e5 + 1.07e6i)14-s − 2.15e6·15-s + (−5.24e5 + 9.08e5i)16-s + (−4.92e6 − 8.52e6i)17-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.635 + 1.10i)5-s + 0.408·6-s + (−0.324 + 0.945i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (0.449 + 0.777i)10-s + (−0.626 − 1.08i)11-s + (0.144 − 0.249i)12-s + 0.607·13-s + (0.464 + 0.533i)14-s − 0.733·15-s + (−0.125 + 0.216i)16-s + (−0.841 − 1.45i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.710 + 0.703i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.710 + 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.244286 - 0.594167i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.244286 - 0.594167i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-16 + 27.7i)T \) |
| 3 | \( 1 + (-121.5 - 210. i)T \) |
| 7 | \( 1 + (1.44e4 - 4.20e4i)T \) |
| good | 5 | \( 1 + (4.43e3 - 7.68e3i)T + (-2.44e7 - 4.22e7i)T^{2} \) |
| 11 | \( 1 + (3.34e5 + 5.79e5i)T + (-1.42e11 + 2.47e11i)T^{2} \) |
| 13 | \( 1 - 8.13e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + (4.92e6 + 8.52e6i)T + (-1.71e13 + 2.96e13i)T^{2} \) |
| 19 | \( 1 + (-6.00e6 + 1.03e7i)T + (-5.82e13 - 1.00e14i)T^{2} \) |
| 23 | \( 1 + (1.42e5 - 2.46e5i)T + (-4.76e14 - 8.25e14i)T^{2} \) |
| 29 | \( 1 - 1.90e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + (9.50e7 + 1.64e8i)T + (-1.27e16 + 2.20e16i)T^{2} \) |
| 37 | \( 1 + (8.41e7 - 1.45e8i)T + (-8.89e16 - 1.54e17i)T^{2} \) |
| 41 | \( 1 + 5.04e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 9.47e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + (1.17e9 - 2.03e9i)T + (-1.23e18 - 2.14e18i)T^{2} \) |
| 53 | \( 1 + (2.19e9 + 3.80e9i)T + (-4.63e18 + 8.02e18i)T^{2} \) |
| 59 | \( 1 + (4.59e9 + 7.95e9i)T + (-1.50e19 + 2.61e19i)T^{2} \) |
| 61 | \( 1 + (4.72e8 - 8.19e8i)T + (-2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (-7.47e9 - 1.29e10i)T + (-6.10e19 + 1.05e20i)T^{2} \) |
| 71 | \( 1 - 1.74e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + (2.73e9 + 4.74e9i)T + (-1.56e20 + 2.71e20i)T^{2} \) |
| 79 | \( 1 + (1.11e10 - 1.93e10i)T + (-3.73e20 - 6.47e20i)T^{2} \) |
| 83 | \( 1 + 7.41e9T + 1.28e21T^{2} \) |
| 89 | \( 1 + (-1.96e9 + 3.39e9i)T + (-1.38e21 - 2.40e21i)T^{2} \) |
| 97 | \( 1 + 1.53e10T + 7.15e21T^{2} \) |
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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
−13.26812677405581015060262287346, −11.53652889968062393492759330034, −11.07535292721323342647403086286, −9.626388514607958077964398097383, −8.376433341212609685861453897850, −6.58026894617902320845379598314, −5.02412671863840142405804245018, −3.26287215813189323506559826026, −2.68907316172342464858589636712, −0.17045994983007705947069161674,
1.41017739815296691818861848691, 3.68599848251949236341184665477, 4.80076956017496878533675226296, 6.55472289773564537949924685916, 7.79023554992473667031767411751, 8.641154664040193408489212265829, 10.34185879651051346165750522505, 12.23565027363616234892682159812, 12.89099826640038999540420198078, 13.87693221800644924690837174135
