| L(s) = 1 | + (−1.22 − 0.707i)2-s + (0.866 + 1.5i)3-s + (0.999 + 1.73i)4-s − 2.44i·6-s + (−4.79 − 5.10i)7-s − 2.82i·8-s + (−1.5 + 2.59i)9-s + (0.919 + 1.59i)11-s + (−1.73 + 2.99i)12-s + 5.40·13-s + (2.26 + 9.63i)14-s + (−2.00 + 3.46i)16-s + (−5.02 − 8.71i)17-s + (3.67 − 2.12i)18-s + (7.96 + 4.59i)19-s + ⋯ |
| L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.288 + 0.5i)3-s + (0.249 + 0.433i)4-s − 0.408i·6-s + (−0.684 − 0.728i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.0835 + 0.144i)11-s + (−0.144 + 0.249i)12-s + 0.415·13-s + (0.161 + 0.688i)14-s + (−0.125 + 0.216i)16-s + (−0.295 − 0.512i)17-s + (0.204 − 0.117i)18-s + (0.418 + 0.241i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.887 - 0.461i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.887 - 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2631065985\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2631065985\) |
| \(L(2)\) |
|
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 | \( 1 + (-0.866 - 1.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (4.79 + 5.10i)T \) |
| good | 11 | \( 1 + (-0.919 - 1.59i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 - 5.40T + 169T^{2} \) |
| 17 | \( 1 + (5.02 + 8.71i)T + (-144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-7.96 - 4.59i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-0.797 - 0.460i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + 12.5T + 841T^{2} \) |
| 31 | \( 1 + (36.1 - 20.8i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-6.30 - 3.64i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 - 52.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 8.12iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (-16.9 + 29.4i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (90.1 - 52.0i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (12.5 - 7.26i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-20.5 - 11.8i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-12.9 + 7.46i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 17.9T + 5.04e3T^{2} \) |
| 73 | \( 1 + (62.1 + 107. i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (40.4 - 70.0i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 154.T + 6.88e3T^{2} \) |
| 89 | \( 1 + (58.9 + 34.0i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 88.1T + 9.40e3T^{2} \) |
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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
−9.925728013406620866255811945759, −9.404363698607308472963665486294, −8.632342759217007911072887742638, −7.63165016247470083905716771462, −6.95625921912252854692323762552, −5.89820250864165579624547813420, −4.61881904276028869010455576855, −3.65976098763533123740246829216, −2.88220188624911934129859795003, −1.41249462227949321224637235427,
0.096132470588843859395191154696, 1.62875792399948880520427698785, 2.73488050383181259603535588038, 3.85869014130374139118095862279, 5.41790079995882986795911045373, 6.10915071703004559257857591483, 6.90898352660502866287122214880, 7.74395175665052857649634252637, 8.640056775766832831497648115923, 9.179148807740309811536260203769
