L(s) = 1 | + (1.42 + 2.44i)2-s + (−3.91 + 6.97i)4-s + 15.7i·5-s + 7i·7-s + (−22.6 + 0.420i)8-s + (−38.5 + 22.5i)10-s + 23.5·11-s − 37.7·13-s + (−17.0 + 10.0i)14-s + (−33.3 − 54.6i)16-s + 31.9i·17-s − 28.8i·19-s + (−110. − 61.7i)20-s + (33.6 + 57.3i)22-s + 50.8·23-s + ⋯ |
L(s) = 1 | + (0.505 + 0.862i)2-s + (−0.489 + 0.872i)4-s + 1.41i·5-s + 0.377i·7-s + (−0.999 + 0.0186i)8-s + (−1.21 + 0.713i)10-s + 0.644·11-s − 0.804·13-s + (−0.326 + 0.191i)14-s + (−0.521 − 0.853i)16-s + 0.455i·17-s − 0.348i·19-s + (−1.23 − 0.690i)20-s + (0.325 + 0.556i)22-s + 0.461·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.902 + 0.429i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.902 + 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.590464430\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.590464430\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.42 - 2.44i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 7iT \) |
good | 5 | \( 1 - 15.7iT - 125T^{2} \) |
| 11 | \( 1 - 23.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 37.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 31.9iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 28.8iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 50.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 205. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 176. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 338.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 77.7iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 463. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 352.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 155. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 761.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 40.2T + 2.26e5T^{2} \) |
| 67 | \( 1 - 838. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 70.1T + 3.57e5T^{2} \) |
| 73 | \( 1 - 726.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 992. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 1.11e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.59e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 1.26e3T + 9.12e5T^{2} \) |
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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
−12.16955070328228762207391558016, −11.38421375244105838601736311819, −10.19232972670068643680616876324, −9.127491385800255173402369046346, −7.955416504372472222662032156891, −6.92364435456622317253145370472, −6.36025562757182506790424055289, −5.07654253718620124510930467448, −3.69086699158161750601291868816, −2.58997822615342027481774248801,
0.53091047434298585754218325559, 1.79207074073296743170844719269, 3.56994140427078625172883513622, 4.71287497674226534094068884572, 5.40828259499931867448741115448, 6.93276936615169854797719413930, 8.474696701403061302008318166238, 9.281128784609251987758180404201, 10.08552558799384951898678767575, 11.27694379610669485328830229721