L(s) = 1 | + (2.21 + 0.333i)5-s + (−2.42 − 1.76i)9-s + (2.62 − 3.61i)13-s + (5.03 − 1.63i)17-s + (4.77 + 1.47i)25-s + (1.77 − 5.45i)29-s + (−5.01 + 6.90i)37-s + (8.65 + 6.28i)41-s + (−4.77 − 4.70i)45-s + 7·49-s + (−11.4 − 3.73i)53-s + (6.73 − 4.89i)61-s + (7.01 − 7.12i)65-s + (−0.447 − 0.616i)73-s + (2.78 + 8.55i)81-s + ⋯ |
L(s) = 1 | + (0.988 + 0.148i)5-s + (−0.809 − 0.587i)9-s + (0.728 − 1.00i)13-s + (1.22 − 0.396i)17-s + (0.955 + 0.294i)25-s + (0.329 − 1.01i)29-s + (−0.824 + 1.13i)37-s + (1.35 + 0.982i)41-s + (−0.712 − 0.701i)45-s + 49-s + (−1.57 − 0.512i)53-s + (0.862 − 0.626i)61-s + (0.870 − 0.883i)65-s + (−0.0524 − 0.0721i)73-s + (0.309 + 0.951i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 + 0.468i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.883 + 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.75981 - 0.437632i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.75981 - 0.437632i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2.21 - 0.333i)T \) |
good | 3 | \( 1 + (2.42 + 1.76i)T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-2.62 + 3.61i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-5.03 + 1.63i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-1.77 + 5.45i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (5.01 - 6.90i)T + (-11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-8.65 - 6.28i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (11.4 + 3.73i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-6.73 + 4.89i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (0.447 + 0.616i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-14.1 + 10.2i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (3.90 + 1.26i)T + (78.4 + 57.0i)T^{2} \) |
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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
−10.09221441783985027525874687301, −9.483601964938143959116035540650, −8.530151606627751658830445666802, −7.74683143624205338658771900243, −6.44563476077526137618003865627, −5.87650856240789132888248195746, −5.06371988852954688184914017899, −3.48748800990758407171795152784, −2.67507874919984967645003872521, −1.04918004465847250553674041552,
1.45682696267584176150521325341, 2.61432559501383260608576636747, 3.89029327914601095722313851634, 5.22639925756812934028126382754, 5.80915911267105538015272359216, 6.74580847845025810388934585066, 7.84755858828802554530273061719, 8.828675842109879088297701993484, 9.322611006371804317246043978537, 10.47883592769720468284431259667