L(s) = 1 | − 1.27i·2-s + 6.37·4-s − 3.19·5-s + (17.8 − 4.90i)7-s − 18.3i·8-s + 4.07i·10-s − 43.9i·11-s + 75.7i·13-s + (−6.26 − 22.7i)14-s + 27.5·16-s + 104.·17-s − 74.7i·19-s − 20.3·20-s − 56.1·22-s − 53.8i·23-s + ⋯ |
L(s) = 1 | − 0.451i·2-s + 0.796·4-s − 0.285·5-s + (0.964 − 0.264i)7-s − 0.810i·8-s + 0.128i·10-s − 1.20i·11-s + 1.61i·13-s + (−0.119 − 0.435i)14-s + 0.430·16-s + 1.49·17-s − 0.902i·19-s − 0.227·20-s − 0.544·22-s − 0.488i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.264 + 0.964i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.264 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.734809926\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.734809926\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-17.8 + 4.90i)T \) |
good | 2 | \( 1 + 1.27iT - 8T^{2} \) |
| 5 | \( 1 + 3.19T + 125T^{2} \) |
| 11 | \( 1 + 43.9iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 75.7iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 104.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 74.7iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 53.8iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 30.4iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 128. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 46.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 426.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 332.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 340.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 235. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 544.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 371. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 106.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 974. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 576. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 21.3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 337.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.02e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 73.3iT - 9.12e5T^{2} \) |
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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
−10.45646391877832203455357024024, −9.342489639442137347900348225040, −8.338234408735248075607214597315, −7.47696125884791385237026212674, −6.61732623179530162939728677540, −5.55060856195017648756251003444, −4.27251439166110793648030206613, −3.28054351755478613332098432686, −1.99224135025807321797305602334, −0.886828064129646753008904304494,
1.33896930069929642112339416847, 2.51024717844231309683524986165, 3.83497539294204158590743148862, 5.35037292397082546122778646240, 5.70660427176299618943353249839, 7.20834685607928663987091893462, 7.79556641410438644396108398496, 8.289589770891375474748755554726, 9.882914783245519170784951507379, 10.42486863786061841285423923438