L(s) = 1 | + (0.909 − 0.909i)5-s − 0.654·7-s + (−13.3 − 13.3i)11-s + (−8.32 − 8.32i)13-s + 3.93·17-s + (−16.8 + 16.8i)19-s + 23.1·23-s + 23.3i·25-s + (35.6 + 35.6i)29-s + 45.5i·31-s + (−0.595 + 0.595i)35-s + (−10.1 + 10.1i)37-s + 28.4i·41-s + (−22.7 − 22.7i)43-s − 10.7i·47-s + ⋯ |
L(s) = 1 | + (0.181 − 0.181i)5-s − 0.0935·7-s + (−1.21 − 1.21i)11-s + (−0.640 − 0.640i)13-s + 0.231·17-s + (−0.889 + 0.889i)19-s + 1.00·23-s + 0.933i·25-s + (1.22 + 1.22i)29-s + 1.46i·31-s + (−0.0170 + 0.0170i)35-s + (−0.274 + 0.274i)37-s + 0.694i·41-s + (−0.528 − 0.528i)43-s − 0.229i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.282 - 0.959i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.282 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7835089554\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7835089554\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.909 + 0.909i)T - 25iT^{2} \) |
| 7 | \( 1 + 0.654T + 49T^{2} \) |
| 11 | \( 1 + (13.3 + 13.3i)T + 121iT^{2} \) |
| 13 | \( 1 + (8.32 + 8.32i)T + 169iT^{2} \) |
| 17 | \( 1 - 3.93T + 289T^{2} \) |
| 19 | \( 1 + (16.8 - 16.8i)T - 361iT^{2} \) |
| 23 | \( 1 - 23.1T + 529T^{2} \) |
| 29 | \( 1 + (-35.6 - 35.6i)T + 841iT^{2} \) |
| 31 | \( 1 - 45.5iT - 961T^{2} \) |
| 37 | \( 1 + (10.1 - 10.1i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 - 28.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (22.7 + 22.7i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + 10.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-41.5 + 41.5i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + (21.0 + 21.0i)T + 3.48e3iT^{2} \) |
| 61 | \( 1 + (-68.7 - 68.7i)T + 3.72e3iT^{2} \) |
| 67 | \( 1 + (67.8 - 67.8i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + 33.3T + 5.04e3T^{2} \) |
| 73 | \( 1 + 18.6iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 6.29iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (72.0 - 72.0i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + 10.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 143.T + 9.40e3T^{2} \) |
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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
−10.13096554029708412346201503151, −8.764192060578990068699580617695, −8.401193558583083139856516608036, −7.41641209827859297280525804073, −6.49624134276436998042146026949, −5.40224852933918195794823178583, −5.00799202097937375209437995137, −3.44024033569862817192874861440, −2.76003117025891406454459785728, −1.21564229173920134007611516096,
0.24001345020686397180331951200, 2.12847497979906925455151600589, 2.74515979821127332617404631542, 4.38850911220633956940604414256, 4.84788414181751930303025754746, 6.07019629530865203059172639102, 6.93147843566354313973680469395, 7.63053513233868545769990733625, 8.522578173092034215759783528028, 9.556738162260519488578715937303