L(s) = 1 | + (0.951 − 0.309i)3-s + 1.74i·7-s + (0.809 − 0.587i)9-s + (−1.87 − 1.36i)11-s + (−2.90 − 3.99i)13-s + (−3.55 − 1.15i)17-s + (0.523 − 1.61i)19-s + (0.537 + 1.65i)21-s + (5.10 − 7.02i)23-s + (0.587 − 0.809i)27-s + (−0.964 − 2.96i)29-s + (2.95 − 9.09i)31-s + (−2.20 − 0.717i)33-s + (3.15 + 4.34i)37-s + (−3.99 − 2.90i)39-s + ⋯ |
L(s) = 1 | + (0.549 − 0.178i)3-s + 0.657i·7-s + (0.269 − 0.195i)9-s + (−0.565 − 0.411i)11-s + (−0.805 − 1.10i)13-s + (−0.862 − 0.280i)17-s + (0.120 − 0.369i)19-s + (0.117 + 0.361i)21-s + (1.06 − 1.46i)23-s + (0.113 − 0.155i)27-s + (−0.179 − 0.551i)29-s + (0.530 − 1.63i)31-s + (−0.384 − 0.124i)33-s + (0.518 + 0.713i)37-s + (−0.639 − 0.464i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0796 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0796 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.575469994\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.575469994\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.951 + 0.309i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 1.74iT - 7T^{2} \) |
| 11 | \( 1 + (1.87 + 1.36i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (2.90 + 3.99i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (3.55 + 1.15i)T + (13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-0.523 + 1.61i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-5.10 + 7.02i)T + (-7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (0.964 + 2.96i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.95 + 9.09i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-3.15 - 4.34i)T + (-11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-7.05 + 5.12i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 2.86iT - 43T^{2} \) |
| 47 | \( 1 + (8.04 - 2.61i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-1.27 + 0.415i)T + (42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-3.54 + 2.57i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-12.4 - 9.01i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (8.77 + 2.85i)T + (54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (-0.00728 - 0.0224i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (0.601 - 0.827i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-0.246 - 0.759i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-2.25 - 0.732i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (3.93 + 2.85i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (3.29 - 1.06i)T + (78.4 - 57.0i)T^{2} \) |
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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
−9.229188767823058571663288614536, −8.430634965240426847741418448760, −7.85237402369333892709393951533, −6.95394275553667120042765087650, −6.01269113007277056871779095074, −5.12922535234505135985920031465, −4.23768217593333879729785975951, −2.66584786698570245451464840298, −2.59722607352814864175542527617, −0.57454193732095069284664111586,
1.52000931281250987202240748550, 2.62749884884871147570755598201, 3.72591136318512672747608366086, 4.57759352561281848638856817102, 5.34073675098151836423542949738, 6.77149316084566064323213587377, 7.20008150297737874993828228050, 8.044153014499063784051691198280, 9.013839873030921516151784274688, 9.567175209251153193497334029920