L(s) = 1 | + (−2.10 + 2.10i)3-s + (4.62 − 4.62i)5-s + 3.04·7-s + 0.156i·9-s + (9.15 + 9.15i)11-s + (5.78 + 5.78i)13-s + 19.4i·15-s + 17.6·17-s + (1.15 − 1.15i)19-s + (−6.41 + 6.41i)21-s − 3.45·23-s − 17.8i·25-s + (−19.2 − 19.2i)27-s + (−12.1 − 12.1i)29-s − 38.5i·31-s + ⋯ |
L(s) = 1 | + (−0.700 + 0.700i)3-s + (0.925 − 0.925i)5-s + 0.435·7-s + 0.0174i·9-s + (0.831 + 0.831i)11-s + (0.444 + 0.444i)13-s + 1.29i·15-s + 1.03·17-s + (0.0606 − 0.0606i)19-s + (−0.305 + 0.305i)21-s − 0.150·23-s − 0.712i·25-s + (−0.713 − 0.713i)27-s + (−0.420 − 0.420i)29-s − 1.24i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.879 - 0.475i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.879 - 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.36032 + 0.343811i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.36032 + 0.343811i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (2.10 - 2.10i)T - 9iT^{2} \) |
| 5 | \( 1 + (-4.62 + 4.62i)T - 25iT^{2} \) |
| 7 | \( 1 - 3.04T + 49T^{2} \) |
| 11 | \( 1 + (-9.15 - 9.15i)T + 121iT^{2} \) |
| 13 | \( 1 + (-5.78 - 5.78i)T + 169iT^{2} \) |
| 17 | \( 1 - 17.6T + 289T^{2} \) |
| 19 | \( 1 + (-1.15 + 1.15i)T - 361iT^{2} \) |
| 23 | \( 1 + 3.45T + 529T^{2} \) |
| 29 | \( 1 + (12.1 + 12.1i)T + 841iT^{2} \) |
| 31 | \( 1 + 38.5iT - 961T^{2} \) |
| 37 | \( 1 + (-0.0972 + 0.0972i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 - 51.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (-1.70 - 1.70i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + 24.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (27.0 - 27.0i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + (19.5 + 19.5i)T + 3.48e3iT^{2} \) |
| 61 | \( 1 + (16.7 + 16.7i)T + 3.72e3iT^{2} \) |
| 67 | \( 1 + (-75.8 + 75.8i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + 134.T + 5.04e3T^{2} \) |
| 73 | \( 1 + 112. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 135. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (74.9 - 74.9i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 31.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 31.5T + 9.40e3T^{2} \) |
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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
−13.15631044524913406929623245655, −12.04605702857958146999359263855, −11.16340869597476135800282252352, −9.874992710115441382414829855254, −9.356206567854527360570921938045, −7.919722320660645743591798496796, −6.20380010533115955991526629571, −5.20086074039277086701014650559, −4.25310878954247218975445004319, −1.63092221800856725671928892897,
1.37200512943531283155049211805, 3.33220617384436717732479646181, 5.58300518907927942472752559811, 6.28367410475653378612314518168, 7.30277907420503591490248588390, 8.786728970198052002445738786765, 10.12011320420431323978726231439, 11.05950358171608345016098527175, 11.92271887836510699790458366540, 12.99089949100186247026584755652