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 + (−0.325 − 6.99i)7-s − 2.82i·8-s + (−1.5 − 2.59i)9-s + (−6.09 + 10.5i)11-s + (−1.73 − 2.99i)12-s − 25.3·13-s + (−5.34 − 8.33i)14-s + (−2.00 − 3.46i)16-s + (−14.3 + 24.9i)17-s + (−3.67 − 2.12i)18-s + (13.9 − 8.03i)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 + (−0.0465 − 0.998i)7-s − 0.353i·8-s + (−0.166 − 0.288i)9-s + (−0.554 + 0.959i)11-s + (−0.144 − 0.249i)12-s − 1.95·13-s + (−0.381 − 0.595i)14-s + (−0.125 − 0.216i)16-s + (−0.846 + 1.46i)17-s + (−0.204 − 0.117i)18-s + (0.732 − 0.422i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.594 - 0.803i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.594 - 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3346695215\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3346695215\) |
\(L(2)\) |
|
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 | \( 1 + (0.325 + 6.99i)T \) |
good | 11 | \( 1 + (6.09 - 10.5i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + 25.3T + 169T^{2} \) |
| 17 | \( 1 + (14.3 - 24.9i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-13.9 + 8.03i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (20.5 - 11.8i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 27.9T + 841T^{2} \) |
| 31 | \( 1 + (-20.0 - 11.5i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-25.1 + 14.5i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + 56.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 7.83iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (-11.3 - 19.7i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (42.0 + 24.2i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-62.3 - 36.0i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-99.2 + 57.3i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (61.0 + 35.2i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 6.41T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-19.7 + 34.2i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (27.3 + 47.4i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 135.T + 6.88e3T^{2} \) |
| 89 | \( 1 + (124. - 72.1i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 78.9T + 9.40e3T^{2} \) |
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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
−9.511974006401320525706187624193, −8.100348918271345974311451801148, −7.33387684004606919179206932860, −6.86255249950098822256765174513, −5.61108425061728037690336932200, −4.63367232819875244578413946515, −3.87372472758927960640152234753, −2.57176689454386634706395561300, −1.74828048640013140881243515512, −0.07087423617563718386888596247,
2.49472734048359810844651445496, 2.85007435270449128672492738002, 4.29547615577183241807333120582, 5.15409494438420412613756245476, 5.70932953372273704776430077988, 6.87549625473394561610974569847, 7.79187304872047068048477228421, 8.503158753680451986562799852079, 9.530020907625263974189635364939, 9.969950835567612292834757066078