L(s) = 1 | + (0.0912 − 1.41i)2-s + (0.707 + 0.707i)3-s + (−1.98 − 0.257i)4-s + (1.32 − 1.80i)5-s + (1.06 − 0.933i)6-s + (−1.86 + 1.86i)7-s + (−0.544 + 2.77i)8-s + 1.00i·9-s + (−2.42 − 2.02i)10-s − 0.728i·11-s + (−1.22 − 1.58i)12-s + (−3.12 + 3.12i)13-s + (2.46 + 2.80i)14-s + (2.20 − 0.342i)15-s + (3.86 + 1.02i)16-s + (1.12 + 1.12i)17-s + ⋯ |
L(s) = 1 | + (0.0645 − 0.997i)2-s + (0.408 + 0.408i)3-s + (−0.991 − 0.128i)4-s + (0.590 − 0.807i)5-s + (0.433 − 0.381i)6-s + (−0.705 + 0.705i)7-s + (−0.192 + 0.981i)8-s + 0.333i·9-s + (−0.767 − 0.641i)10-s − 0.219i·11-s + (−0.352 − 0.457i)12-s + (−0.866 + 0.866i)13-s + (0.658 + 0.749i)14-s + (0.570 − 0.0885i)15-s + (0.966 + 0.255i)16-s + (0.272 + 0.272i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.546 + 0.837i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.546 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.817228 - 0.442446i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.817228 - 0.442446i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.0912 + 1.41i)T \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 + (-1.32 + 1.80i)T \) |
good | 7 | \( 1 + (1.86 - 1.86i)T - 7iT^{2} \) |
| 11 | \( 1 + 0.728iT - 11T^{2} \) |
| 13 | \( 1 + (3.12 - 3.12i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1.12 - 1.12i)T + 17iT^{2} \) |
| 19 | \( 1 - 3.73T + 19T^{2} \) |
| 23 | \( 1 + (5.83 + 5.83i)T + 23iT^{2} \) |
| 29 | \( 1 + 2.64iT - 29T^{2} \) |
| 31 | \( 1 + 6.01iT - 31T^{2} \) |
| 37 | \( 1 + (-3.12 - 3.12i)T + 37iT^{2} \) |
| 41 | \( 1 + 4.24T + 41T^{2} \) |
| 43 | \( 1 + (-5.10 - 5.10i)T + 43iT^{2} \) |
| 47 | \( 1 + (-2.09 + 2.09i)T - 47iT^{2} \) |
| 53 | \( 1 + (-0.484 + 0.484i)T - 53iT^{2} \) |
| 59 | \( 1 - 4.92T + 59T^{2} \) |
| 61 | \( 1 - 2.31T + 61T^{2} \) |
| 67 | \( 1 + (5.10 - 5.10i)T - 67iT^{2} \) |
| 71 | \( 1 - 13.1iT - 71T^{2} \) |
| 73 | \( 1 + (-3.96 + 3.96i)T - 73iT^{2} \) |
| 79 | \( 1 + 7.11T + 79T^{2} \) |
| 83 | \( 1 + (-3.55 - 3.55i)T + 83iT^{2} \) |
| 89 | \( 1 + 1.03iT - 89T^{2} \) |
| 97 | \( 1 + (12.5 + 12.5i)T + 97iT^{2} \) |
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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
−14.61475605772709772827622503276, −13.66455272458973499926049386285, −12.60571447260429012613425372687, −11.74302201500702253236755527125, −9.978912696771910400214745553378, −9.456896137864312631782028417290, −8.347483012465051057578954155923, −5.80193236186153991877192141527, −4.35002774704167954760368500884, −2.45183576347053094532963864216,
3.37130257409424508583511936349, 5.57950822027377763077666186519, 6.96363737989232728965577444062, 7.67344560209680778661792403648, 9.474051717135374842054123744816, 10.23931636111382693776655762026, 12.34747804584616398179878238383, 13.52446230718183543050247813021, 14.12514395374100963679250615448, 15.16712187851486138980539456248