L(s) = 1 | + (−0.707 + 0.707i)2-s + (−1.65 + 0.516i)3-s − 1.00i·4-s + (−0.633 − 2.14i)5-s + (0.803 − 1.53i)6-s + (−2.44 − 2.44i)7-s + (0.707 + 0.707i)8-s + (2.46 − 1.70i)9-s + (1.96 + 1.06i)10-s + 6.26i·11-s + (0.516 + 1.65i)12-s + (2.06 − 2.06i)13-s + 3.46·14-s + (2.15 + 3.21i)15-s − 1.00·16-s + (5.14 − 5.14i)17-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + (−0.954 + 0.298i)3-s − 0.500i·4-s + (−0.283 − 0.959i)5-s + (0.327 − 0.626i)6-s + (−0.925 − 0.925i)7-s + (0.250 + 0.250i)8-s + (0.821 − 0.569i)9-s + (0.621 + 0.337i)10-s + 1.89i·11-s + (0.149 + 0.477i)12-s + (0.572 − 0.572i)13-s + 0.925·14-s + (0.556 + 0.830i)15-s − 0.250·16-s + (1.24 − 1.24i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.936 + 0.350i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.936 + 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0304014 - 0.167789i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0304014 - 0.167789i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.65 - 0.516i)T \) |
| 5 | \( 1 + (0.633 + 2.14i)T \) |
| 31 | \( 1 - T \) |
good | 7 | \( 1 + (2.44 + 2.44i)T + 7iT^{2} \) |
| 11 | \( 1 - 6.26iT - 11T^{2} \) |
| 13 | \( 1 + (-2.06 + 2.06i)T - 13iT^{2} \) |
| 17 | \( 1 + (-5.14 + 5.14i)T - 17iT^{2} \) |
| 19 | \( 1 + 3.95iT - 19T^{2} \) |
| 23 | \( 1 + (-0.977 - 0.977i)T + 23iT^{2} \) |
| 29 | \( 1 + 7.01T + 29T^{2} \) |
| 37 | \( 1 + (6.92 + 6.92i)T + 37iT^{2} \) |
| 41 | \( 1 - 7.39iT - 41T^{2} \) |
| 43 | \( 1 + (1.60 - 1.60i)T - 43iT^{2} \) |
| 47 | \( 1 + (7.18 - 7.18i)T - 47iT^{2} \) |
| 53 | \( 1 + (4.50 + 4.50i)T + 53iT^{2} \) |
| 59 | \( 1 - 1.19T + 59T^{2} \) |
| 61 | \( 1 + 7.46T + 61T^{2} \) |
| 67 | \( 1 + (4.48 + 4.48i)T + 67iT^{2} \) |
| 71 | \( 1 + 3.67iT - 71T^{2} \) |
| 73 | \( 1 + (1.34 - 1.34i)T - 73iT^{2} \) |
| 79 | \( 1 - 2.94iT - 79T^{2} \) |
| 83 | \( 1 + (7.44 + 7.44i)T + 83iT^{2} \) |
| 89 | \( 1 - 3.81T + 89T^{2} \) |
| 97 | \( 1 + (-5.39 - 5.39i)T + 97iT^{2} \) |
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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
−9.618097264252741241971270110028, −9.250628943002984134734175863745, −7.62992916812896356635032669442, −7.30382848323987417716081755232, −6.35099258032244610062894505716, −5.21238574488738831082964497097, −4.68023252520095825104613269380, −3.55239307033345910230677447144, −1.30984115261547624101986083902, −0.11929646127957065340805748583,
1.63918068049459190470089254495, 3.19285202119393426955065430974, 3.72759102668098988508161065466, 5.73994706214597335682967909422, 6.01746773552534943455004736034, 6.92294621542175144235883978042, 8.055820972391785140334440206519, 8.721436741225045417622543416841, 9.905524436419519474882915525502, 10.53150074440693011537549690154