L(s) = 1 | + (0.707 + 0.707i)2-s + (1.41 − i)3-s + 1.00i·4-s + (0.707 − 0.707i)5-s + (1.70 + 0.292i)6-s + 2·7-s + (−0.707 + 0.707i)8-s + (1.00 − 2.82i)9-s + 1.00·10-s + (1.00 + 1.41i)12-s + (1 + i)13-s + (1.41 + 1.41i)14-s + (0.292 − 1.70i)15-s − 1.00·16-s + (4.24 − 4.24i)17-s + (2.70 − 1.29i)18-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + (0.816 − 0.577i)3-s + 0.500i·4-s + (0.316 − 0.316i)5-s + (0.696 + 0.119i)6-s + 0.755·7-s + (−0.250 + 0.250i)8-s + (0.333 − 0.942i)9-s + 0.316·10-s + (0.288 + 0.408i)12-s + (0.277 + 0.277i)13-s + (0.377 + 0.377i)14-s + (0.0756 − 0.440i)15-s − 0.250·16-s + (1.02 − 1.02i)17-s + (0.638 − 0.304i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.209956801\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.209956801\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 3 | \( 1 + (-1.41 + i)T \) |
| 5 | \( 1 + (-0.707 + 0.707i)T \) |
| 37 | \( 1 + (1 + 6i)T \) |
good | 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + (-1 - i)T + 13iT^{2} \) |
| 17 | \( 1 + (-4.24 + 4.24i)T - 17iT^{2} \) |
| 19 | \( 1 + (3 + 3i)T + 19iT^{2} \) |
| 23 | \( 1 + (2.82 - 2.82i)T - 23iT^{2} \) |
| 29 | \( 1 + (-4.24 - 4.24i)T + 29iT^{2} \) |
| 31 | \( 1 + (7 - 7i)T - 31iT^{2} \) |
| 41 | \( 1 - 5.65T + 41T^{2} \) |
| 43 | \( 1 + (-7 - 7i)T + 43iT^{2} \) |
| 47 | \( 1 - 2.82iT - 47T^{2} \) |
| 53 | \( 1 + 5.65iT - 53T^{2} \) |
| 59 | \( 1 + (-5.65 + 5.65i)T - 59iT^{2} \) |
| 61 | \( 1 + (9 - 9i)T - 61iT^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 2.82iT - 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 + (-3 - 3i)T + 79iT^{2} \) |
| 83 | \( 1 + 2.82iT - 83T^{2} \) |
| 89 | \( 1 + (1.41 + 1.41i)T + 89iT^{2} \) |
| 97 | \( 1 + (-7 - 7i)T + 97iT^{2} \) |
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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
−9.423012086922539346659598510929, −8.929753602367932166235388604958, −8.024705037590234877826023504104, −7.40891472771879649146324224441, −6.58754683602131486024868992050, −5.55620948851054911034164030240, −4.69407833192904972852518071414, −3.61885296998006644548725224888, −2.53304099209962704796509945757, −1.32688143756791562871071688207,
1.65005308206818863920438839104, 2.55415962484326028358638140084, 3.72602480215257132495847754023, 4.32228864671638402418084076242, 5.47929088624892800989972403653, 6.20775614029697427791815305532, 7.67200813477317898392618582805, 8.210991429022719152470729493684, 9.136899735804455290838653728209, 10.14116991560248139664009266079