L(s) = 1 | + (0.125 − 1.99i)2-s + (−3.96 − 0.5i)4-s + (−4.94 + 4.94i)7-s + (−1.49 + 7.85i)8-s + (6.36 + 6.36i)9-s + (−15.9 + 6.60i)11-s + (9.26 + 10.5i)14-s + (15.5 + 3.96i)16-s + (13.5 − 11.9i)18-s + (11.1 + 32.6i)22-s + (12.1 + 12.1i)23-s + (−17.6 + 17.6i)25-s + (22.1 − 17.1i)28-s + (−1.02 − 0.424i)29-s + (9.86 − 30.4i)32-s + ⋯ |
L(s) = 1 | + (0.0626 − 0.998i)2-s + (−0.992 − 0.125i)4-s + (−0.707 + 0.707i)7-s + (−0.186 + 0.982i)8-s + (0.707 + 0.707i)9-s + (−1.44 + 0.600i)11-s + (0.661 + 0.750i)14-s + (0.968 + 0.248i)16-s + (0.750 − 0.661i)18-s + (0.508 + 1.48i)22-s + (0.528 + 0.528i)23-s + (−0.707 + 0.707i)25-s + (0.789 − 0.613i)28-s + (−0.0353 − 0.0146i)29-s + (0.308 − 0.951i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.487 - 0.872i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.487 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.648894 + 0.380689i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.648894 + 0.380689i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.125 + 1.99i)T \) |
| 7 | \( 1 + (4.94 - 4.94i)T \) |
good | 3 | \( 1 + (-6.36 - 6.36i)T^{2} \) |
| 5 | \( 1 + (17.6 - 17.6i)T^{2} \) |
| 11 | \( 1 + (15.9 - 6.60i)T + (85.5 - 85.5i)T^{2} \) |
| 13 | \( 1 + (119. + 119. i)T^{2} \) |
| 17 | \( 1 + 289T^{2} \) |
| 19 | \( 1 + (255. + 255. i)T^{2} \) |
| 23 | \( 1 + (-12.1 - 12.1i)T + 529iT^{2} \) |
| 29 | \( 1 + (1.02 + 0.424i)T + (594. + 594. i)T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 + (-4.13 - 9.98i)T + (-968. + 968. i)T^{2} \) |
| 41 | \( 1 - 1.68e3iT^{2} \) |
| 43 | \( 1 + (71.9 - 29.8i)T + (1.30e3 - 1.30e3i)T^{2} \) |
| 47 | \( 1 + 2.20e3T^{2} \) |
| 53 | \( 1 + (-42.5 + 17.6i)T + (1.98e3 - 1.98e3i)T^{2} \) |
| 59 | \( 1 + (2.46e3 - 2.46e3i)T^{2} \) |
| 61 | \( 1 + (-2.63e3 - 2.63e3i)T^{2} \) |
| 67 | \( 1 + (78.2 + 32.4i)T + (3.17e3 + 3.17e3i)T^{2} \) |
| 71 | \( 1 + (59.8 - 59.8i)T - 5.04e3iT^{2} \) |
| 73 | \( 1 - 5.32e3iT^{2} \) |
| 79 | \( 1 - 23.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (4.87e3 + 4.87e3i)T^{2} \) |
| 89 | \( 1 + 7.92e3iT^{2} \) |
| 97 | \( 1 - 9.40e3T^{2} \) |
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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
−12.24887919947692616509894018939, −11.21670413430389434035646709861, −10.17943306257337627746410008208, −9.660027849655339342706564361692, −8.415151474246194817512860635872, −7.34192557039458790907914437520, −5.61539814466611701251673560671, −4.71807783612912724964773627436, −3.16097539221147259327103074883, −1.97881038624140451074088826742,
0.39628387631228295883303581626, 3.30547028396814748049010646527, 4.50614807743334454131192218755, 5.81870930452265543576691763452, 6.81205442811634421860088437260, 7.67439292378974322935058394773, 8.745618039077534390583994723315, 9.897987526494636909810255760086, 10.52394983674406398763184555506, 12.19553317253624494142732979385