| L(s) = 1 | + (−0.290 − 1.08i)2-s + (−0.130 − 0.991i)3-s + (0.639 − 0.369i)4-s + (0.501 + 1.87i)5-s + (−1.03 + 0.429i)6-s + (2.20 − 1.45i)7-s + (−2.17 − 2.17i)8-s + (−0.965 + 0.258i)9-s + (1.88 − 1.08i)10-s + (0.134 + 1.02i)11-s + (−0.449 − 0.586i)12-s + (−4.60 − 1.90i)13-s + (−2.22 − 1.97i)14-s + (1.79 − 0.742i)15-s + (−0.988 + 1.71i)16-s + (1.73 + 1.32i)17-s + ⋯ |
| L(s) = 1 | + (−0.205 − 0.767i)2-s + (−0.0753 − 0.572i)3-s + (0.319 − 0.184i)4-s + (0.224 + 0.837i)5-s + (−0.423 + 0.175i)6-s + (0.835 − 0.549i)7-s + (−0.768 − 0.768i)8-s + (−0.321 + 0.0862i)9-s + (0.596 − 0.344i)10-s + (0.0406 + 0.308i)11-s + (−0.129 − 0.169i)12-s + (−1.27 − 0.528i)13-s + (−0.593 − 0.527i)14-s + (0.462 − 0.191i)15-s + (−0.247 + 0.427i)16-s + (0.420 + 0.322i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.785 + 0.618i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.785 + 0.618i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.490734 - 1.41678i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.490734 - 1.41678i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.130 + 0.991i)T \) |
| 7 | \( 1 + (-2.20 + 1.45i)T \) |
| 41 | \( 1 + (-4.98 + 4.02i)T \) |
| good | 2 | \( 1 + (0.290 + 1.08i)T + (-1.73 + i)T^{2} \) |
| 5 | \( 1 + (-0.501 - 1.87i)T + (-4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (-0.134 - 1.02i)T + (-10.6 + 2.84i)T^{2} \) |
| 13 | \( 1 + (4.60 + 1.90i)T + (9.19 + 9.19i)T^{2} \) |
| 17 | \( 1 + (-1.73 - 1.32i)T + (4.39 + 16.4i)T^{2} \) |
| 19 | \( 1 + (-0.829 + 6.30i)T + (-18.3 - 4.91i)T^{2} \) |
| 23 | \( 1 + (2.32 + 1.34i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.86 + 6.91i)T + (-20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + (1.12 + 1.94i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.323 - 0.560i)T + (-18.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (-6.45 + 6.45i)T - 43iT^{2} \) |
| 47 | \( 1 + (1.65 - 12.5i)T + (-45.3 - 12.1i)T^{2} \) |
| 53 | \( 1 + (0.211 + 1.60i)T + (-51.1 + 13.7i)T^{2} \) |
| 59 | \( 1 + (-10.0 + 5.81i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (13.6 - 3.64i)T + (52.8 - 30.5i)T^{2} \) |
| 67 | \( 1 + (0.582 - 0.758i)T + (-17.3 - 64.7i)T^{2} \) |
| 71 | \( 1 + (3.54 - 8.55i)T + (-50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (1.29 - 4.83i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-6.16 + 4.73i)T + (20.4 - 76.3i)T^{2} \) |
| 83 | \( 1 - 15.4iT - 83T^{2} \) |
| 89 | \( 1 + (5.89 - 4.52i)T + (23.0 - 85.9i)T^{2} \) |
| 97 | \( 1 + (-3.49 - 8.44i)T + (-68.5 + 68.5i)T^{2} \) |
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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
−10.11607240378180614875889050007, −9.287821480824563125657137076381, −7.935834596591303712148801022340, −7.25963773664047524323119309110, −6.56849099260347918933965300748, −5.51119029677325350538402154959, −4.28350376379077564110870608290, −2.77056735922103029344674099301, −2.23505313083243039529264938766, −0.77220708916325697366659176425,
1.76269385780377915555446484693, 3.09239337823888532104901496422, 4.58567355180583632059123165481, 5.33087914045072508191966150331, 5.98347065264874757322272503290, 7.25208615284689051083483384152, 7.989798202760452794978271810394, 8.766137080614050470816109663117, 9.375438004602036268098072272993, 10.37846130489538389077489638391
