L(s) = 1 | + (1.35 − 1.47i)2-s + (−0.350 − 3.98i)4-s + 10.9i·7-s + (−6.35 − 4.86i)8-s − 11.3i·11-s − 10.8·13-s + (16.1 + 14.7i)14-s + (−15.7 + 2.79i)16-s − 15.8·17-s + 24.9i·19-s + (−16.7 − 15.3i)22-s + 20.9i·23-s + (−14.5 + 15.9i)26-s + (43.5 − 3.83i)28-s − 22.8·29-s + ⋯ |
L(s) = 1 | + (0.675 − 0.737i)2-s + (−0.0876 − 0.996i)4-s + 1.55i·7-s + (−0.793 − 0.608i)8-s − 1.03i·11-s − 0.831·13-s + (1.15 + 1.05i)14-s + (−0.984 + 0.174i)16-s − 0.929·17-s + 1.31i·19-s + (−0.761 − 0.697i)22-s + 0.911i·23-s + (−0.561 + 0.613i)26-s + (1.55 − 0.136i)28-s − 0.786·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0876 - 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0876 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6985127399\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6985127399\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.35 + 1.47i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 10.9iT - 49T^{2} \) |
| 11 | \( 1 + 11.3iT - 121T^{2} \) |
| 13 | \( 1 + 10.8T + 169T^{2} \) |
| 17 | \( 1 + 15.8T + 289T^{2} \) |
| 19 | \( 1 - 24.9iT - 361T^{2} \) |
| 23 | \( 1 - 20.9iT - 529T^{2} \) |
| 29 | \( 1 + 22.8T + 841T^{2} \) |
| 31 | \( 1 + 22.7iT - 961T^{2} \) |
| 37 | \( 1 + 19.1T + 1.36e3T^{2} \) |
| 41 | \( 1 + 17T + 1.68e3T^{2} \) |
| 43 | \( 1 - 6.51iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 38.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 13.2T + 2.80e3T^{2} \) |
| 59 | \( 1 - 95.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 92.0T + 3.72e3T^{2} \) |
| 67 | \( 1 - 54.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 68.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 44.1T + 5.32e3T^{2} \) |
| 79 | \( 1 + 81.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 27.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 42.1T + 7.92e3T^{2} \) |
| 97 | \( 1 - 154.T + 9.40e3T^{2} \) |
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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
−10.23190233503081071237903361108, −9.303306162578511376322075881380, −8.798195697214764417183563472445, −7.64310836701570980159401025520, −6.18417806231336820251277395199, −5.75638171654694553603062133063, −4.86533644235862988177321484337, −3.63600184556765564603502359295, −2.67906678990703307554251980351, −1.73861561880539784939769040706,
0.16459720036137021684057819183, 2.22227904354351068322235291464, 3.55507707422690274208430481141, 4.59259430510740993080177222119, 4.92862177822437517757439413291, 6.55709512869970238916668453327, 7.03433670030860530347853457160, 7.61029958217588109238355087278, 8.711966776705164994658765473004, 9.637779290897989907423547841399