| L(s) = 1 | + (1.22 − 0.707i)2-s + (−1.12 − 2.78i)3-s + (0.999 − 1.73i)4-s + (−0.422 + 0.244i)5-s + (−3.34 − 2.60i)6-s + (4.69 + 5.19i)7-s − 2.82i·8-s + (−6.46 + 6.26i)9-s + (−0.345 + 0.597i)10-s + (13.1 + 7.58i)11-s + (−5.94 − 0.830i)12-s − 17.3·13-s + (9.41 + 3.04i)14-s + (1.15 + 0.900i)15-s + (−2.00 − 3.46i)16-s + (−0.422 − 0.244i)17-s + ⋯ |
| L(s) = 1 | + (0.612 − 0.353i)2-s + (−0.375 − 0.926i)3-s + (0.249 − 0.433i)4-s + (−0.0845 + 0.0488i)5-s + (−0.557 − 0.434i)6-s + (0.670 + 0.742i)7-s − 0.353i·8-s + (−0.718 + 0.695i)9-s + (−0.0345 + 0.0597i)10-s + (1.19 + 0.689i)11-s + (−0.495 − 0.0692i)12-s − 1.33·13-s + (0.672 + 0.217i)14-s + (0.0769 + 0.0600i)15-s + (−0.125 − 0.216i)16-s + (−0.0248 − 0.0143i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.507 + 0.861i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.507 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.15325 - 0.659357i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.15325 - 0.659357i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.22 + 0.707i)T \) |
| 3 | \( 1 + (1.12 + 2.78i)T \) |
| 7 | \( 1 + (-4.69 - 5.19i)T \) |
| good | 5 | \( 1 + (0.422 - 0.244i)T + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (-13.1 - 7.58i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + 17.3T + 169T^{2} \) |
| 17 | \( 1 + (0.422 + 0.244i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (6.53 + 11.3i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (5.78 - 3.34i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 47.3iT - 841T^{2} \) |
| 31 | \( 1 + (-14.2 + 24.6i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 - 28.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 2.14T + 1.84e3T^{2} \) |
| 47 | \( 1 + (63.7 - 36.7i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (52.7 + 30.4i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-87.4 - 50.4i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-17.1 - 29.6i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-49.9 + 86.5i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 82.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-25.8 + 44.8i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-33.3 - 57.7i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 88.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-50.6 + 29.2i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 - 25.0T + 9.40e3T^{2} \) |
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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
−15.18949767273521941723914820837, −14.40667752393682801401613751621, −13.08789388675677944310866447252, −11.94273499914336133587184334767, −11.48110790909442719022978109686, −9.552852705957949517555118063061, −7.73429435121119324281430506172, −6.32911699900054416417586711327, −4.79927998329660573689296949336, −2.12843528403547952178587957365,
3.83322646708001216848073302714, 5.07098090224306135603626642584, 6.71414358444992867473273150842, 8.456798366678759535000110216796, 10.06916878864108101664922999296, 11.32457783777955269243351111691, 12.31313473899030655222871125201, 14.26809426474816761934694851339, 14.54050651823651327051485689534, 16.06693776296084427255282127388
