| L(s) = 1 | + (0.966 + 1.03i)2-s + (−1.15 − 0.229i)3-s + (−0.131 + 1.99i)4-s + (−1.56 + 1.59i)5-s + (−0.878 − 1.41i)6-s + (0.0938 + 0.0388i)7-s + (−2.18 + 1.79i)8-s + (−1.49 − 0.617i)9-s + (−3.16 − 0.0745i)10-s + (−0.0544 − 0.0364i)11-s + (0.609 − 2.27i)12-s + (−2.30 + 1.54i)13-s + (0.0506 + 0.134i)14-s + (2.17 − 1.48i)15-s + (−3.96 − 0.522i)16-s + 2.67i·17-s + ⋯ |
| L(s) = 1 | + (0.683 + 0.729i)2-s + (−0.666 − 0.132i)3-s + (−0.0655 + 0.997i)4-s + (−0.700 + 0.713i)5-s + (−0.358 − 0.577i)6-s + (0.0354 + 0.0146i)7-s + (−0.773 + 0.634i)8-s + (−0.497 − 0.205i)9-s + (−0.999 − 0.0235i)10-s + (−0.0164 − 0.0109i)11-s + (0.175 − 0.656i)12-s + (−0.639 + 0.427i)13-s + (0.0135 + 0.0359i)14-s + (0.561 − 0.382i)15-s + (−0.991 − 0.130i)16-s + 0.648i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.124i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 - 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0535132 + 0.856728i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0535132 + 0.856728i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.966 - 1.03i)T \) |
| 5 | \( 1 + (1.56 - 1.59i)T \) |
| good | 3 | \( 1 + (1.15 + 0.229i)T + (2.77 + 1.14i)T^{2} \) |
| 7 | \( 1 + (-0.0938 - 0.0388i)T + (4.94 + 4.94i)T^{2} \) |
| 11 | \( 1 + (0.0544 + 0.0364i)T + (4.20 + 10.1i)T^{2} \) |
| 13 | \( 1 + (2.30 - 1.54i)T + (4.97 - 12.0i)T^{2} \) |
| 17 | \( 1 - 2.67iT - 17T^{2} \) |
| 19 | \( 1 + (-0.0181 + 0.0910i)T + (-17.5 - 7.27i)T^{2} \) |
| 23 | \( 1 + (-1.44 - 3.48i)T + (-16.2 + 16.2i)T^{2} \) |
| 29 | \( 1 + (3.80 - 2.54i)T + (11.0 - 26.7i)T^{2} \) |
| 31 | \( 1 - 5.49T + 31T^{2} \) |
| 37 | \( 1 + (4.29 - 6.43i)T + (-14.1 - 34.1i)T^{2} \) |
| 41 | \( 1 + (-6.08 - 2.52i)T + (28.9 + 28.9i)T^{2} \) |
| 43 | \( 1 + (-0.207 - 1.04i)T + (-39.7 + 16.4i)T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 + (-1.38 - 6.95i)T + (-48.9 + 20.2i)T^{2} \) |
| 59 | \( 1 + (-5.59 + 1.11i)T + (54.5 - 22.5i)T^{2} \) |
| 61 | \( 1 + (-3.72 + 2.49i)T + (23.3 - 56.3i)T^{2} \) |
| 67 | \( 1 + (0.687 - 3.45i)T + (-61.8 - 25.6i)T^{2} \) |
| 71 | \( 1 + (7.82 - 3.24i)T + (50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (1.32 + 3.20i)T + (-51.6 + 51.6i)T^{2} \) |
| 79 | \( 1 + (-8.52 + 8.52i)T - 79iT^{2} \) |
| 83 | \( 1 + (7.11 - 4.75i)T + (31.7 - 76.6i)T^{2} \) |
| 89 | \( 1 + (4.37 + 10.5i)T + (-62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 + (4.98 - 4.98i)T - 97iT^{2} \) |
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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
−11.93997388087482532308663311590, −11.54931159958005572008442992332, −10.51237722339356584993436122724, −9.032778438060468779880011151140, −7.965423770214363153182664384517, −7.05588948866985114082209086174, −6.28300757198310578661961295504, −5.25796058136833308817129361950, −4.06034549333907277529977717187, −2.87554379401044902776282999811,
0.52149219790015399004074303888, 2.63097891088295773944188361492, 4.11848723330643291625330705217, 5.04317978531954416249906193073, 5.78005042036537347627323896534, 7.20337265066590087255688858701, 8.500996250945402241919695580025, 9.532419828897386134786055777039, 10.62835232475141054602704699539, 11.33479800720309511921597593764
