L(s) = 1 | + (1.00 − 1.20i)2-s + (0.781 + 2.14i)3-s + (−0.0809 − 0.459i)4-s + (3.37 + 1.22i)6-s + (−3.47 + 2.00i)7-s + (2.08 + 1.20i)8-s + (−1.70 + 1.42i)9-s + (−1.38 + 2.39i)11-s + (0.922 − 0.532i)12-s + (−0.953 + 2.61i)13-s + (−1.09 + 6.20i)14-s + (4.43 − 1.61i)16-s + (2.31 − 2.76i)17-s + 3.48i·18-s + (−1.79 − 3.97i)19-s + ⋯ |
L(s) = 1 | + (0.713 − 0.850i)2-s + (0.451 + 1.23i)3-s + (−0.0404 − 0.229i)4-s + (1.37 + 0.501i)6-s + (−1.31 + 0.758i)7-s + (0.737 + 0.425i)8-s + (−0.567 + 0.475i)9-s + (−0.417 + 0.722i)11-s + (0.266 − 0.153i)12-s + (−0.264 + 0.726i)13-s + (−0.292 + 1.65i)14-s + (1.10 − 0.403i)16-s + (0.562 − 0.670i)17-s + 0.822i·18-s + (−0.410 − 0.911i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.521 - 0.853i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.521 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.83351 + 1.02812i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.83351 + 1.02812i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 + (1.79 + 3.97i)T \) |
good | 2 | \( 1 + (-1.00 + 1.20i)T + (-0.347 - 1.96i)T^{2} \) |
| 3 | \( 1 + (-0.781 - 2.14i)T + (-2.29 + 1.92i)T^{2} \) |
| 7 | \( 1 + (3.47 - 2.00i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.38 - 2.39i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.953 - 2.61i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-2.31 + 2.76i)T + (-2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-1.34 + 0.237i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-7.28 + 6.11i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (0.776 + 1.34i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 8.51iT - 37T^{2} \) |
| 41 | \( 1 + (-6.21 + 2.26i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-6.15 - 1.08i)T + (40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-5.85 - 6.97i)T + (-8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (3.88 - 0.684i)T + (49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (2.76 + 2.32i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (1.30 + 7.37i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (9.36 + 11.1i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (0.576 - 3.26i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (3.42 + 9.40i)T + (-55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (1.82 - 0.666i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (1.40 - 0.809i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (11.5 + 4.19i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-7.45 + 8.87i)T + (-16.8 - 95.5i)T^{2} \) |
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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
−11.16019379387680939799886703031, −10.18783873880723743760871040440, −9.598842677276188817039666947685, −8.920947013171522383695612168539, −7.56477099170191311089184522219, −6.28886134360694988381304481757, −4.87461428198346350522813973462, −4.31928788542062430274394197964, −3.07288941588577122933325399628, −2.56006202678697269731408124076,
1.04174660092930070537128564153, 2.91076880172974505140348136091, 4.00034445576419417667784889464, 5.60824820886222779529172369506, 6.25897510137036097793348110017, 7.15349225284940448935415460617, 7.67961442988209921060026433798, 8.692985351261252606696963058418, 10.20037628908328967093954008603, 10.59090709176005081671502294632