L(s) = 1 | + (−1.18 + 0.771i)2-s + (−0.713 + 1.57i)3-s + (0.808 − 1.82i)4-s + (−0.373 − 2.42i)6-s + (1.44 − 1.44i)7-s + (0.454 + 2.79i)8-s + (−1.98 − 2.25i)9-s − 0.641·11-s + (2.31 + 2.58i)12-s + (−2.03 + 2.03i)13-s + (−0.597 + 2.83i)14-s + (−2.69 − 2.95i)16-s + (4.37 + 4.37i)17-s + (4.08 + 1.13i)18-s + 4.93·19-s + ⋯ |
L(s) = 1 | + (−0.837 + 0.545i)2-s + (−0.411 + 0.911i)3-s + (0.404 − 0.914i)4-s + (−0.152 − 0.988i)6-s + (0.546 − 0.546i)7-s + (0.160 + 0.987i)8-s + (−0.660 − 0.750i)9-s − 0.193·11-s + (0.667 + 0.744i)12-s + (−0.563 + 0.563i)13-s + (−0.159 + 0.756i)14-s + (−0.673 − 0.739i)16-s + (1.06 + 1.06i)17-s + (0.963 + 0.267i)18-s + 1.13·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.337933 + 0.684451i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.337933 + 0.684451i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.18 - 0.771i)T \) |
| 3 | \( 1 + (0.713 - 1.57i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-1.44 + 1.44i)T - 7iT^{2} \) |
| 11 | \( 1 + 0.641T + 11T^{2} \) |
| 13 | \( 1 + (2.03 - 2.03i)T - 13iT^{2} \) |
| 17 | \( 1 + (-4.37 - 4.37i)T + 17iT^{2} \) |
| 19 | \( 1 - 4.93T + 19T^{2} \) |
| 23 | \( 1 + (3.73 - 3.73i)T - 23iT^{2} \) |
| 29 | \( 1 - 9.84iT - 29T^{2} \) |
| 31 | \( 1 - 5.23T + 31T^{2} \) |
| 37 | \( 1 + (3.44 + 3.44i)T + 37iT^{2} \) |
| 41 | \( 1 - 7.20iT - 41T^{2} \) |
| 43 | \( 1 + (4.37 - 4.37i)T - 43iT^{2} \) |
| 47 | \( 1 + (4.08 + 4.08i)T + 47iT^{2} \) |
| 53 | \( 1 + (3.83 + 3.83i)T + 53iT^{2} \) |
| 59 | \( 1 + 2.50iT - 59T^{2} \) |
| 61 | \( 1 - 9.64iT - 61T^{2} \) |
| 67 | \( 1 + (2.59 + 2.59i)T + 67iT^{2} \) |
| 71 | \( 1 + 16.7iT - 71T^{2} \) |
| 73 | \( 1 + (-8.40 - 8.40i)T + 73iT^{2} \) |
| 79 | \( 1 - 5.31iT - 79T^{2} \) |
| 83 | \( 1 + (-2.20 - 2.20i)T + 83iT^{2} \) |
| 89 | \( 1 - 3.96T + 89T^{2} \) |
| 97 | \( 1 + (1.11 - 1.11i)T - 97iT^{2} \) |
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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
−10.70072850825035736168939085276, −10.00596887019663250892076947921, −9.422801115012557570620162526010, −8.323780813193599043346799331941, −7.58060699462671057921815619458, −6.51612859325352973405013547884, −5.47278133429323342891990497797, −4.75674298176205414946151890520, −3.37972888396975367689346154958, −1.40084394594113568039876482172,
0.63946366062939612657197817199, 2.10458970998954451881143411042, 3.05335237750847854934003281696, 4.88918163249188192662647496142, 5.89727044329944014242133392168, 7.09466888747597281180262950259, 7.86345112883045301953803383940, 8.367486908369380084386009924563, 9.647983519038996219857624042263, 10.29039538964228957875226619901