| 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
−13.87693221800644924690837174135, −12.89099826640038999540420198078, −12.23565027363616234892682159812, −10.34185879651051346165750522505, −8.641154664040193408489212265829, −7.79023554992473667031767411751, −6.55472289773564537949924685916, −4.80076956017496878533675226296, −3.68599848251949236341184665477, −1.41017739815296691818861848691,
0.17045994983007705947069161674, 2.68907316172342464858589636712, 3.26287215813189323506559826026, 5.02412671863840142405804245018, 6.58026894617902320845379598314, 8.376433341212609685861453897850, 9.626388514607958077964398097383, 11.07535292721323342647403086286, 11.53652889968062393492759330034, 13.26812677405581015060262287346
