L(s) = 1 | − 3.41i·3-s − 0.558·7-s − 2.66·9-s + (2.82 + 10.6i)11-s + 18.1·13-s − 32.0·17-s + 15.7i·19-s + 1.90i·21-s + 31.7i·23-s − 21.6i·27-s + 48.3i·29-s − 43.3·31-s + (36.3 − 9.63i)33-s − 21.4i·37-s − 61.8i·39-s + ⋯ |
L(s) = 1 | − 1.13i·3-s − 0.0798·7-s − 0.296·9-s + (0.256 + 0.966i)11-s + 1.39·13-s − 1.88·17-s + 0.828i·19-s + 0.0908i·21-s + 1.38i·23-s − 0.801i·27-s + 1.66i·29-s − 1.39·31-s + (1.10 − 0.291i)33-s − 0.579i·37-s − 1.58i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.661i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.473006906\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.473006906\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + (-2.82 - 10.6i)T \) |
good | 3 | \( 1 + 3.41iT - 9T^{2} \) |
| 7 | \( 1 + 0.558T + 49T^{2} \) |
| 13 | \( 1 - 18.1T + 169T^{2} \) |
| 17 | \( 1 + 32.0T + 289T^{2} \) |
| 19 | \( 1 - 15.7iT - 361T^{2} \) |
| 23 | \( 1 - 31.7iT - 529T^{2} \) |
| 29 | \( 1 - 48.3iT - 841T^{2} \) |
| 31 | \( 1 + 43.3T + 961T^{2} \) |
| 37 | \( 1 + 21.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 38.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 23.3T + 1.84e3T^{2} \) |
| 47 | \( 1 - 75.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 8.43iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 29.6T + 3.48e3T^{2} \) |
| 61 | \( 1 - 64.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 18.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 94.8T + 5.04e3T^{2} \) |
| 73 | \( 1 + 0.945T + 5.32e3T^{2} \) |
| 79 | \( 1 + 73.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 161.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 91.5T + 7.92e3T^{2} \) |
| 97 | \( 1 - 33.1iT - 9.40e3T^{2} \) |
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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
−9.560688157828573407548545661098, −8.907541336364549836279924125317, −7.961815926947536275999591161689, −7.17234166084346612543687895377, −6.57416250225053056014008256323, −5.75025095021812207368598259793, −4.47824656954555316107862507049, −3.49291460791156765569386920525, −1.98885811453913987723278672653, −1.34883420016209913067147313763,
0.46001767808590404955280995440, 2.25880309800253444278491085233, 3.60033405286759396849506276167, 4.16854241124014197872785708289, 5.11310217729925274393144554940, 6.20462836038497579821021963114, 6.82332798633900909959358490411, 8.307884371766754594545507487600, 8.841060217441537735191548738055, 9.454520365101088149533144819061