| L(s) = 1 | + (−27.7 − 16i)2-s + (−76.1 − 413. i)3-s + (511. + 886. i)4-s + (−4.60e3 + 7.97e3i)5-s + (−4.51e3 + 1.26e4i)6-s + (−2.61e4 − 3.59e4i)7-s − 3.27e4i·8-s + (−1.65e5 + 6.30e4i)9-s + (2.55e5 − 1.47e5i)10-s + (−7.27e5 + 4.20e5i)11-s + (3.28e5 − 2.79e5i)12-s − 1.15e6i·13-s + (1.48e5 + 1.41e6i)14-s + (3.65e6 + 1.29e6i)15-s + (−5.24e5 + 9.08e5i)16-s + (5.51e6 + 9.55e6i)17-s + ⋯ |
| L(s) = 1 | + (−0.612 − 0.353i)2-s + (−0.180 − 0.983i)3-s + (0.249 + 0.433i)4-s + (−0.658 + 1.14i)5-s + (−0.236 + 0.666i)6-s + (−0.587 − 0.809i)7-s − 0.353i·8-s + (−0.934 + 0.355i)9-s + (0.806 − 0.465i)10-s + (−1.36 + 0.786i)11-s + (0.380 − 0.324i)12-s − 0.862i·13-s + (0.0738 + 0.703i)14-s + (1.24 + 0.441i)15-s + (−0.125 + 0.216i)16-s + (0.942 + 1.63i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.631 + 0.775i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.631 + 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.578044 - 0.274717i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.578044 - 0.274717i\) |
| \(L(\frac{13}{2})\) |
|
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 | \( 1 + (76.1 + 413. i)T \) |
| 7 | \( 1 + (2.61e4 + 3.59e4i)T \) |
| good | 5 | \( 1 + (4.60e3 - 7.97e3i)T + (-2.44e7 - 4.22e7i)T^{2} \) |
| 11 | \( 1 + (7.27e5 - 4.20e5i)T + (1.42e11 - 2.47e11i)T^{2} \) |
| 13 | \( 1 + 1.15e6iT - 1.79e12T^{2} \) |
| 17 | \( 1 + (-5.51e6 - 9.55e6i)T + (-1.71e13 + 2.96e13i)T^{2} \) |
| 19 | \( 1 + (-2.84e6 - 1.64e6i)T + (5.82e13 + 1.00e14i)T^{2} \) |
| 23 | \( 1 + (3.04e7 + 1.75e7i)T + (4.76e14 + 8.25e14i)T^{2} \) |
| 29 | \( 1 + 8.82e7iT - 1.22e16T^{2} \) |
| 31 | \( 1 + (2.70e7 - 1.56e7i)T + (1.27e16 - 2.20e16i)T^{2} \) |
| 37 | \( 1 + (-2.80e8 + 4.85e8i)T + (-8.89e16 - 1.54e17i)T^{2} \) |
| 41 | \( 1 - 6.02e7T + 5.50e17T^{2} \) |
| 43 | \( 1 - 7.98e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + (-1.32e9 + 2.29e9i)T + (-1.23e18 - 2.14e18i)T^{2} \) |
| 53 | \( 1 + (3.07e9 - 1.77e9i)T + (4.63e18 - 8.02e18i)T^{2} \) |
| 59 | \( 1 + (2.88e9 + 4.98e9i)T + (-1.50e19 + 2.61e19i)T^{2} \) |
| 61 | \( 1 + (-1.76e8 - 1.02e8i)T + (2.17e19 + 3.76e19i)T^{2} \) |
| 67 | \( 1 + (-4.17e9 - 7.22e9i)T + (-6.10e19 + 1.05e20i)T^{2} \) |
| 71 | \( 1 + 5.34e9iT - 2.31e20T^{2} \) |
| 73 | \( 1 + (1.30e10 - 7.55e9i)T + (1.56e20 - 2.71e20i)T^{2} \) |
| 79 | \( 1 + (-1.53e10 + 2.66e10i)T + (-3.73e20 - 6.47e20i)T^{2} \) |
| 83 | \( 1 - 5.81e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + (-4.18e9 + 7.25e9i)T + (-1.38e21 - 2.40e21i)T^{2} \) |
| 97 | \( 1 - 1.54e10iT - 7.15e21T^{2} \) |
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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
−13.06692335996635723690807018290, −12.28050759419382014230790969492, −10.74455736041660564232462962434, −10.25583401344027126814147723741, −7.86572512772568904980646815361, −7.52845781703456268629559757240, −6.10309641847282257512066383720, −3.54734816428749604060728159199, −2.30312069227854591971246224969, −0.49990943311355728149954210055,
0.54887344078265241559550494469, 2.99093560005215914030658566370, 4.82615882851481679602718755991, 5.77229632949608581158690491669, 7.86132936119762334182994235887, 8.969949278677379420726732804858, 9.746249478889129625488138832024, 11.30444778906148886744117086162, 12.26788063917098070524920678053, 13.93837614940460203006211553927
