L(s) = 1 | + 1.29i·5-s + i·7-s + 2.21·11-s − 1.03·13-s + 2.24i·17-s − 8.63i·19-s − 8.96·23-s + 3.31·25-s − 3.26i·29-s − 4.37i·31-s − 1.29·35-s − 7.21·37-s + 7.58i·41-s + 12.9i·43-s + 5.24·47-s + ⋯ |
L(s) = 1 | + 0.579i·5-s + 0.377i·7-s + 0.668·11-s − 0.287·13-s + 0.544i·17-s − 1.98i·19-s − 1.87·23-s + 0.663·25-s − 0.606i·29-s − 0.786i·31-s − 0.219·35-s − 1.18·37-s + 1.18i·41-s + 1.97i·43-s + 0.764·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3702224982\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3702224982\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 5 | \( 1 - 1.29iT - 5T^{2} \) |
| 11 | \( 1 - 2.21T + 11T^{2} \) |
| 13 | \( 1 + 1.03T + 13T^{2} \) |
| 17 | \( 1 - 2.24iT - 17T^{2} \) |
| 19 | \( 1 + 8.63iT - 19T^{2} \) |
| 23 | \( 1 + 8.96T + 23T^{2} \) |
| 29 | \( 1 + 3.26iT - 29T^{2} \) |
| 31 | \( 1 + 4.37iT - 31T^{2} \) |
| 37 | \( 1 + 7.21T + 37T^{2} \) |
| 41 | \( 1 - 7.58iT - 41T^{2} \) |
| 43 | \( 1 - 12.9iT - 43T^{2} \) |
| 47 | \( 1 - 5.24T + 47T^{2} \) |
| 53 | \( 1 + 11.1iT - 53T^{2} \) |
| 59 | \( 1 + 2.46T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 - 4.46iT - 67T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 + 12.8T + 73T^{2} \) |
| 79 | \( 1 + 9.14iT - 79T^{2} \) |
| 83 | \( 1 + 9.96T + 83T^{2} \) |
| 89 | \( 1 + 3.48iT - 89T^{2} \) |
| 97 | \( 1 + 6.30T + 97T^{2} \) |
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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
−7.75243726894612986445984640446, −7.07453963117820437290956159500, −6.32147823908891304759881344587, −5.91964699063883194998549111258, −4.76204398846779873231164725098, −4.25352024874740784815551661995, −3.18586076785750390816434074231, −2.53066727161536138261165622270, −1.56379560688370656229154166118, −0.091510857620809956142561656216,
1.27111877920716736440291527206, 1.98897830281905938938721592325, 3.30255104068881651260923813735, 3.96527524043900104827764165326, 4.64046811047949697294948604178, 5.60782214853045053437342401831, 6.03708364672826464619880250107, 7.15544794508325632210044718934, 7.46352516454879161088853755254, 8.578971027197203215894820556953