L(s) = 1 | + (−2.18 + 1.58i)2-s + (−0.867 − 2.66i)3-s + (1.64 − 5.04i)4-s + (0.360 + 0.261i)5-s + (6.13 + 4.46i)6-s + (0.309 − 0.951i)7-s + (2.76 + 8.50i)8-s + (−3.94 + 2.86i)9-s − 1.20·10-s − 14.8·12-s + (−0.364 + 0.265i)13-s + (0.835 + 2.57i)14-s + (0.386 − 1.18i)15-s + (−10.9 − 7.96i)16-s + (−3.90 − 2.83i)17-s + (4.07 − 12.5i)18-s + ⋯ |
L(s) = 1 | + (−1.54 + 1.12i)2-s + (−0.500 − 1.54i)3-s + (0.820 − 2.52i)4-s + (0.161 + 0.116i)5-s + (2.50 + 1.82i)6-s + (0.116 − 0.359i)7-s + (0.976 + 3.00i)8-s + (−1.31 + 0.956i)9-s − 0.380·10-s − 4.30·12-s + (−0.101 + 0.0735i)13-s + (0.223 + 0.687i)14-s + (0.0996 − 0.306i)15-s + (−2.74 − 1.99i)16-s + (−0.947 − 0.688i)17-s + (0.960 − 2.95i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 - 0.220i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 - 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0119274 + 0.106765i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0119274 + 0.106765i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-0.309 + 0.951i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (2.18 - 1.58i)T + (0.618 - 1.90i)T^{2} \) |
| 3 | \( 1 + (0.867 + 2.66i)T + (-2.42 + 1.76i)T^{2} \) |
| 5 | \( 1 + (-0.360 - 0.261i)T + (1.54 + 4.75i)T^{2} \) |
| 13 | \( 1 + (0.364 - 0.265i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (3.90 + 2.83i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (0.335 + 1.03i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 - 4.57T + 23T^{2} \) |
| 29 | \( 1 + (0.613 - 1.88i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (6.68 - 4.85i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-2.26 + 6.95i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-0.547 - 1.68i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 11.4T + 43T^{2} \) |
| 47 | \( 1 + (-0.315 - 0.972i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-2.88 + 2.09i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (4.44 - 13.6i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (3.98 + 2.89i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + 6.18T + 67T^{2} \) |
| 71 | \( 1 + (-4.79 - 3.48i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (0.512 - 1.57i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-2.91 + 2.12i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-8.74 - 6.35i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + 5.21T + 89T^{2} \) |
| 97 | \( 1 + (-4.29 + 3.12i)T + (29.9 - 92.2i)T^{2} \) |
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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
−9.399660686567797316123575428726, −8.695636377293794960527189887480, −7.83926821075408554467919124377, −7.05940431457099959134521029403, −6.79264069964254595546140329042, −5.89057041973755322014256755823, −4.97302947168628427261292999987, −2.33753180529087762257487355049, −1.28538910638404859907114673898, −0.098087580831008067910494081516,
1.82399271158008973093450752573, 3.16143583100785863126398729849, 4.01317440718989665728604339604, 5.10408801146832155102252453588, 6.40581333938970891297195613819, 7.68701345278742351665040647582, 8.679318726601041797592490525572, 9.219142982429777080705374515399, 9.832246588601086656458665220670, 10.53053405472021168249724587979