| L(s) = 1 | + (−27.7 − 16i)2-s + (−296. − 298. i)3-s + (511. + 886. i)4-s + (−3.22e3 + 5.57e3i)5-s + (3.42e3 + 1.30e4i)6-s + (2.53e4 + 3.65e4i)7-s − 3.27e4i·8-s + (−1.57e3 + 1.77e5i)9-s + (1.78e5 − 1.03e5i)10-s + (1.04e5 − 6.04e4i)11-s + (1.13e5 − 4.15e5i)12-s − 5.09e4i·13-s + (−1.17e5 − 1.41e6i)14-s + (2.62e6 − 6.90e5i)15-s + (−5.24e5 + 9.08e5i)16-s + (−4.53e6 − 7.85e6i)17-s + ⋯ |
| L(s) = 1 | + (−0.612 − 0.353i)2-s + (−0.703 − 0.710i)3-s + (0.249 + 0.433i)4-s + (−0.460 + 0.798i)5-s + (0.179 + 0.683i)6-s + (0.569 + 0.821i)7-s − 0.353i·8-s + (−0.00890 + 0.999i)9-s + (0.564 − 0.325i)10-s + (0.196 − 0.113i)11-s + (0.131 − 0.482i)12-s − 0.0380i·13-s + (−0.0585 − 0.704i)14-s + (0.891 − 0.234i)15-s + (−0.125 + 0.216i)16-s + (−0.774 − 1.34i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 - 0.305i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.952 - 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.0183397 + 0.117314i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0183397 + 0.117314i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (27.7 + 16i)T \) |
| 3 | \( 1 + (296. + 298. i)T \) |
| 7 | \( 1 + (-2.53e4 - 3.65e4i)T \) |
| good | 5 | \( 1 + (3.22e3 - 5.57e3i)T + (-2.44e7 - 4.22e7i)T^{2} \) |
| 11 | \( 1 + (-1.04e5 + 6.04e4i)T + (1.42e11 - 2.47e11i)T^{2} \) |
| 13 | \( 1 + 5.09e4iT - 1.79e12T^{2} \) |
| 17 | \( 1 + (4.53e6 + 7.85e6i)T + (-1.71e13 + 2.96e13i)T^{2} \) |
| 19 | \( 1 + (-8.06e6 - 4.65e6i)T + (5.82e13 + 1.00e14i)T^{2} \) |
| 23 | \( 1 + (-2.27e7 - 1.31e7i)T + (4.76e14 + 8.25e14i)T^{2} \) |
| 29 | \( 1 - 6.87e6iT - 1.22e16T^{2} \) |
| 31 | \( 1 + (2.29e8 - 1.32e8i)T + (1.27e16 - 2.20e16i)T^{2} \) |
| 37 | \( 1 + (-1.04e8 + 1.81e8i)T + (-8.89e16 - 1.54e17i)T^{2} \) |
| 41 | \( 1 - 1.44e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.24e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + (-7.02e7 + 1.21e8i)T + (-1.23e18 - 2.14e18i)T^{2} \) |
| 53 | \( 1 + (4.50e9 - 2.59e9i)T + (4.63e18 - 8.02e18i)T^{2} \) |
| 59 | \( 1 + (2.19e9 + 3.79e9i)T + (-1.50e19 + 2.61e19i)T^{2} \) |
| 61 | \( 1 + (2.62e9 + 1.51e9i)T + (2.17e19 + 3.76e19i)T^{2} \) |
| 67 | \( 1 + (-4.30e9 - 7.46e9i)T + (-6.10e19 + 1.05e20i)T^{2} \) |
| 71 | \( 1 + 1.02e10iT - 2.31e20T^{2} \) |
| 73 | \( 1 + (-2.57e10 + 1.48e10i)T + (1.56e20 - 2.71e20i)T^{2} \) |
| 79 | \( 1 + (-1.06e10 + 1.84e10i)T + (-3.73e20 - 6.47e20i)T^{2} \) |
| 83 | \( 1 + 3.90e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + (4.37e10 - 7.57e10i)T + (-1.38e21 - 2.40e21i)T^{2} \) |
| 97 | \( 1 + 9.26e10iT - 7.15e21T^{2} \) |
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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
−14.02331941265022885167117524177, −12.52985207727502744158547705320, −11.46479852362148153586460290176, −10.98757759795586391388371946044, −9.217197971952738720557105113180, −7.75320619737264943333969588448, −6.81149264469663912140437560105, −5.19879795386642061638757712744, −2.96193663665666565767230472837, −1.55560699800745792361767022880,
0.05516585177050729196228030024, 1.24124002076431000384389355323, 4.03833627506909135508565788246, 5.09324906639239461566967929870, 6.68700906144239877144788085821, 8.169860160769508306133549512927, 9.337383277187530703737046875053, 10.65009231066372131500088347872, 11.48735632666691078309828359174, 12.90868261476385998285276220253
