| L(s) = 1 | + (−1.58 + 1.22i)2-s + (1.01 − 3.86i)4-s + 8.58i·5-s + 1.97i·7-s + (3.11 + 7.36i)8-s + (−10.4 − 13.5i)10-s − 19.5·11-s − 18.4i·13-s + (−2.41 − 3.12i)14-s + (−13.9 − 7.86i)16-s − 22.2i·17-s + (18.8 − 2.33i)19-s + (33.1 + 8.72i)20-s + (30.9 − 23.8i)22-s − 15.3·23-s + ⋯ |
| L(s) = 1 | + (−0.791 + 0.610i)2-s + (0.254 − 0.967i)4-s + 1.71i·5-s + 0.282i·7-s + (0.389 + 0.921i)8-s + (−1.04 − 1.35i)10-s − 1.77·11-s − 1.41i·13-s + (−0.172 − 0.223i)14-s + (−0.870 − 0.491i)16-s − 1.30i·17-s + (0.992 − 0.122i)19-s + (1.65 + 0.436i)20-s + (1.40 − 1.08i)22-s − 0.665·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.732 + 0.681i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.732 + 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5070719171\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5070719171\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.58 - 1.22i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-18.8 + 2.33i)T \) |
| good | 5 | \( 1 - 8.58iT - 25T^{2} \) |
| 7 | \( 1 - 1.97iT - 49T^{2} \) |
| 11 | \( 1 + 19.5T + 121T^{2} \) |
| 13 | \( 1 + 18.4iT - 169T^{2} \) |
| 17 | \( 1 + 22.2iT - 289T^{2} \) |
| 23 | \( 1 + 15.3T + 529T^{2} \) |
| 29 | \( 1 - 38.6T + 841T^{2} \) |
| 31 | \( 1 + 45.2T + 961T^{2} \) |
| 37 | \( 1 + 9.92iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 36.5T + 1.68e3T^{2} \) |
| 43 | \( 1 - 25.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 6.62T + 2.20e3T^{2} \) |
| 53 | \( 1 + 68.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + 83.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 14.9T + 3.72e3T^{2} \) |
| 67 | \( 1 - 92.3T + 4.48e3T^{2} \) |
| 71 | \( 1 + 43.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 45.8T + 5.32e3T^{2} \) |
| 79 | \( 1 + 84.8T + 6.24e3T^{2} \) |
| 83 | \( 1 + 11.0T + 6.88e3T^{2} \) |
| 89 | \( 1 + 58.4T + 7.92e3T^{2} \) |
| 97 | \( 1 + 30.6iT - 9.40e3T^{2} \) |
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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
−10.15185702193132994049228179903, −9.523030724403287574555871073277, −8.049222719942662809026820353394, −7.64034468756128872064975511221, −6.88535721496204512910905193608, −5.77415727499868277164444715786, −5.16584229508481944960521170444, −3.04673110900892945051939948467, −2.48656476715447203153693455251, −0.25752389861644335885709916156,
1.14709439210738597011877407143, 2.22137475565052937017181724358, 3.84739043869145300025619203787, 4.70678928287044314987600615694, 5.77754077838442323831990943571, 7.26503800588622859901929311084, 8.106911449799792481630388933358, 8.660459238162122008310615435733, 9.510637570709917235494407805770, 10.24584170779706358851652173704
