L(s) = 1 | + (1 + i)3-s + (−2 − i)5-s − i·9-s + (3 + 3i)11-s + (3 + 3i)13-s + (−1 − 3i)15-s + 4i·17-s + (1 − i)19-s + 8·23-s + (3 + 4i)25-s + (4 − 4i)27-s + (−3 + 3i)29-s + 6i·33-s + (3 − 3i)37-s + 6i·39-s + ⋯ |
L(s) = 1 | + (0.577 + 0.577i)3-s + (−0.894 − 0.447i)5-s − 0.333i·9-s + (0.904 + 0.904i)11-s + (0.832 + 0.832i)13-s + (−0.258 − 0.774i)15-s + 0.970i·17-s + (0.229 − 0.229i)19-s + 1.66·23-s + (0.600 + 0.800i)25-s + (0.769 − 0.769i)27-s + (−0.557 + 0.557i)29-s + 1.04i·33-s + (0.493 − 0.493i)37-s + 0.960i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.56679 + 0.584787i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.56679 + 0.584787i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (2 + i)T \) |
good | 3 | \( 1 + (-1 - i)T + 3iT^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + (-3 - 3i)T + 11iT^{2} \) |
| 13 | \( 1 + (-3 - 3i)T + 13iT^{2} \) |
| 17 | \( 1 - 4iT - 17T^{2} \) |
| 19 | \( 1 + (-1 + i)T - 19iT^{2} \) |
| 23 | \( 1 - 8T + 23T^{2} \) |
| 29 | \( 1 + (3 - 3i)T - 29iT^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + (-3 + 3i)T - 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + (-3 + 3i)T - 43iT^{2} \) |
| 47 | \( 1 - 2iT - 47T^{2} \) |
| 53 | \( 1 + (9 - 9i)T - 53iT^{2} \) |
| 59 | \( 1 + (9 + 9i)T + 59iT^{2} \) |
| 61 | \( 1 + (-5 + 5i)T - 61iT^{2} \) |
| 67 | \( 1 + (3 + 3i)T + 67iT^{2} \) |
| 71 | \( 1 - 6iT - 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + (-9 - 9i)T + 83iT^{2} \) |
| 89 | \( 1 + 12iT - 89T^{2} \) |
| 97 | \( 1 + 12iT - 97T^{2} \) |
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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
−10.78172020231951529349331618719, −9.309052643929400939831162097620, −9.212166991515026729153209364727, −8.249970181903206552212641775091, −7.17640594601093678381061456742, −6.33155384156427002838438079982, −4.78058021721825793901118436806, −4.05003879570420221979853626278, −3.29607478800013542968253112356, −1.41871590420643330443648040446,
1.05048749606800997945248718505, 2.85932301323980646345280206241, 3.51571458567721468512967394391, 4.87897275117825871014168310986, 6.17062625916564443362799696570, 7.10730270760551130815197086871, 7.87340508267125837874704573501, 8.534225383247845858711106321751, 9.403595301214600236728374904902, 10.79233743332433850022592320559