L(s) = 1 | + (2.83 − 0.968i)3-s + 5.58i·5-s + 3.90·7-s + (7.12 − 5.50i)9-s + 14.3i·11-s + 3.60·13-s + (5.41 + 15.8i)15-s − 26.6i·17-s − 2.16·19-s + (11.0 − 3.78i)21-s − 17.4i·23-s − 6.22·25-s + (14.8 − 22.5i)27-s + 43.2i·29-s − 24.0·31-s + ⋯ |
L(s) = 1 | + (0.946 − 0.322i)3-s + 1.11i·5-s + 0.558·7-s + (0.791 − 0.611i)9-s + 1.30i·11-s + 0.277·13-s + (0.360 + 1.05i)15-s − 1.56i·17-s − 0.113·19-s + (0.528 − 0.180i)21-s − 0.760i·23-s − 0.248·25-s + (0.551 − 0.833i)27-s + 1.49i·29-s − 0.775·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 - 0.322i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.946 - 0.322i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.01845 + 0.334821i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.01845 + 0.334821i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.83 + 0.968i)T \) |
| 13 | \( 1 - 3.60T \) |
good | 5 | \( 1 - 5.58iT - 25T^{2} \) |
| 7 | \( 1 - 3.90T + 49T^{2} \) |
| 11 | \( 1 - 14.3iT - 121T^{2} \) |
| 17 | \( 1 + 26.6iT - 289T^{2} \) |
| 19 | \( 1 + 2.16T + 361T^{2} \) |
| 23 | \( 1 + 17.4iT - 529T^{2} \) |
| 29 | \( 1 - 43.2iT - 841T^{2} \) |
| 31 | \( 1 + 24.0T + 961T^{2} \) |
| 37 | \( 1 - 3.55T + 1.36e3T^{2} \) |
| 41 | \( 1 + 48.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 84.6T + 1.84e3T^{2} \) |
| 47 | \( 1 + 33.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 9.24iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 36.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 23.8T + 3.72e3T^{2} \) |
| 67 | \( 1 + 112.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 93.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 24.0T + 5.32e3T^{2} \) |
| 79 | \( 1 - 44.4T + 6.24e3T^{2} \) |
| 83 | \( 1 + 55.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 77.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 143.T + 9.40e3T^{2} \) |
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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
−12.85010147269953070627142544101, −11.81303865094395443453145120653, −10.64197630808215967235668611170, −9.687486097001715559053342756437, −8.582363730057201148824588242272, −7.27354106240636384552261015637, −6.86782372196175223130635857424, −4.83828271616498087636157538117, −3.28112725012474563853108264995, −2.03184231875343350088683558312,
1.54773376826465004485970458968, 3.50816157990504524445919634394, 4.68060911486806638461520940367, 6.00215582219601171136643932961, 7.964001683239004599992539796496, 8.426827840488541946324302016149, 9.334814336008754998753726635844, 10.56140600364036634149886540000, 11.63496411157588879806073842658, 13.01215614011283288589260352071