L(s) = 1 | + (−1.22 − 1.22i)2-s + (1.22 + 1.22i)3-s + 0.999i·4-s − 2.99i·6-s + (2.44 − 2.44i)7-s + (−1.22 + 1.22i)8-s + 2.99i·9-s − i·11-s + (−1.22 + 1.22i)12-s + (−2.44 − 2.44i)13-s − 5.99·14-s + 5·16-s + (−4.89 − 4.89i)17-s + (3.67 − 3.67i)18-s − 2i·19-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.866i)2-s + (0.707 + 0.707i)3-s + 0.499i·4-s − 1.22i·6-s + (0.925 − 0.925i)7-s + (−0.433 + 0.433i)8-s + 0.999i·9-s − 0.301i·11-s + (−0.353 + 0.353i)12-s + (−0.679 − 0.679i)13-s − 1.60·14-s + 1.25·16-s + (−1.18 − 1.18i)17-s + (0.866 − 0.866i)18-s − 0.458i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.727223 - 0.918885i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.727223 - 0.918885i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.22 - 1.22i)T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + iT \) |
good | 2 | \( 1 + (1.22 + 1.22i)T + 2iT^{2} \) |
| 7 | \( 1 + (-2.44 + 2.44i)T - 7iT^{2} \) |
| 13 | \( 1 + (2.44 + 2.44i)T + 13iT^{2} \) |
| 17 | \( 1 + (4.89 + 4.89i)T + 17iT^{2} \) |
| 19 | \( 1 + 2iT - 19T^{2} \) |
| 23 | \( 1 + (-4.89 + 4.89i)T - 23iT^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 37iT^{2} \) |
| 41 | \( 1 - 6iT - 41T^{2} \) |
| 43 | \( 1 + (7.34 + 7.34i)T + 43iT^{2} \) |
| 47 | \( 1 + (-4.89 - 4.89i)T + 47iT^{2} \) |
| 53 | \( 1 + (-4.89 + 4.89i)T - 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + (-2.44 + 2.44i)T - 67iT^{2} \) |
| 71 | \( 1 + 12iT - 71T^{2} \) |
| 73 | \( 1 + (-2.44 - 2.44i)T + 73iT^{2} \) |
| 79 | \( 1 + 10iT - 79T^{2} \) |
| 83 | \( 1 + (-7.34 + 7.34i)T - 83iT^{2} \) |
| 89 | \( 1 + 12T + 89T^{2} \) |
| 97 | \( 1 + (9.79 - 9.79i)T - 97iT^{2} \) |
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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
−10.09619104549368042180291106231, −9.216164044524615144838718227161, −8.542007611832232955798072121079, −7.85545033730745535570406563494, −6.82666626883813141003240335385, −5.06622347074096381439117967567, −4.55857720227933559484928522294, −3.07371749957662860629297308751, −2.31535271266320038460972321147, −0.73640372214755439074320240339,
1.54326258745598864247748837231, 2.64628511012115663194687078614, 4.12988969494698604803712393759, 5.48082224056076259347807263938, 6.57353562060503103428983826745, 7.12103515215724364093797038254, 8.138357489388960078870660168080, 8.536837510662383344235487078022, 9.192219503324366506916536985152, 10.04580446556114442705416211324