L(s) = 1 | + (−0.258 − 0.965i)2-s + (−1.61 + 0.431i)3-s + (−0.866 + 0.499i)4-s + (0.834 + 1.44i)6-s + (2.81 + 2.81i)7-s + (0.707 + 0.707i)8-s + (−0.185 + 0.107i)9-s − 5.57·11-s + (1.18 − 1.18i)12-s + (−0.831 + 3.10i)13-s + (1.98 − 3.44i)14-s + (0.500 − 0.866i)16-s + (3.97 − 1.06i)17-s + (0.151 + 0.151i)18-s + (0.254 − 4.35i)19-s + ⋯ |
L(s) = 1 | + (−0.183 − 0.683i)2-s + (−0.930 + 0.249i)3-s + (−0.433 + 0.249i)4-s + (0.340 + 0.590i)6-s + (1.06 + 1.06i)7-s + (0.249 + 0.249i)8-s + (−0.0618 + 0.0356i)9-s − 1.68·11-s + (0.340 − 0.340i)12-s + (−0.230 + 0.861i)13-s + (0.531 − 0.919i)14-s + (0.125 − 0.216i)16-s + (0.963 − 0.258i)17-s + (0.0356 + 0.0356i)18-s + (0.0584 − 0.998i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.000356651 + 0.00776635i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.000356651 + 0.00776635i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.258 + 0.965i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-0.254 + 4.35i)T \) |
good | 3 | \( 1 + (1.61 - 0.431i)T + (2.59 - 1.5i)T^{2} \) |
| 7 | \( 1 + (-2.81 - 2.81i)T + 7iT^{2} \) |
| 11 | \( 1 + 5.57T + 11T^{2} \) |
| 13 | \( 1 + (0.831 - 3.10i)T + (-11.2 - 6.5i)T^{2} \) |
| 17 | \( 1 + (-3.97 + 1.06i)T + (14.7 - 8.5i)T^{2} \) |
| 23 | \( 1 + (3.83 + 1.02i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (2.09 + 3.62i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 0.573iT - 31T^{2} \) |
| 37 | \( 1 + (7.24 - 7.24i)T - 37iT^{2} \) |
| 41 | \( 1 + (5.12 + 2.96i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.44 + 5.37i)T + (-37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (-3.15 + 11.7i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (0.981 - 3.66i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (3.75 - 6.50i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.06 + 1.83i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (11.4 + 3.07i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (4.56 + 2.63i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.50 - 13.0i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (2.48 - 4.30i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (7.30 - 7.30i)T - 83iT^{2} \) |
| 89 | \( 1 + (7.08 + 12.2i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.619 + 2.31i)T + (-84.0 + 48.5i)T^{2} \) |
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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
−10.51068924566085529769817058097, −9.929498912743697630293451983609, −8.724072669741431745727789663225, −8.205468431075852548933993776396, −7.16667158879245527279706079080, −5.67075870498977265029155611448, −5.24924734908576745981019642647, −4.50244774968535408806109937508, −2.84200350774215283557080747374, −1.95796512792318329100639640126,
0.00442666981907145145925466865, 1.41582702646282966862717232482, 3.33959652634949478591765243673, 4.72320044737226925148715680482, 5.42958123651171790316450447771, 5.99086724796532713205529519540, 7.34655284271920193966473486366, 7.75733952520693771533439914491, 8.386377454563515576075485727306, 9.875610625040932025714183650370