L(s) = 1 | + 27i·3-s − 64·4-s − 286i·7-s − 729·9-s − 1.72e3i·12-s − 506i·13-s + 4.09e3·16-s + 1.05e4·19-s + 7.72e3·21-s − 1.96e4i·27-s + 1.83e4i·28-s + 3.52e4·31-s + 4.66e4·36-s − 8.92e4i·37-s + 1.36e4·39-s + ⋯ |
L(s) = 1 | + i·3-s − 4-s − 0.833i·7-s − 0.999·9-s − i·12-s − 0.230i·13-s + 16-s + 1.54·19-s + 0.833·21-s − 0.999i·27-s + 0.833i·28-s + 1.18·31-s + 0.999·36-s − 1.76i·37-s + 0.230·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(1.12414 - 0.265374i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.12414 - 0.265374i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 27iT \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + 64T^{2} \) |
| 7 | \( 1 + 286iT - 1.17e5T^{2} \) |
| 11 | \( 1 - 1.77e6T^{2} \) |
| 13 | \( 1 + 506iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 2.41e7T^{2} \) |
| 19 | \( 1 - 1.05e4T + 4.70e7T^{2} \) |
| 23 | \( 1 + 1.48e8T^{2} \) |
| 29 | \( 1 - 5.94e8T^{2} \) |
| 31 | \( 1 - 3.52e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 8.92e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 4.75e9T^{2} \) |
| 43 | \( 1 + 1.11e5iT - 6.32e9T^{2} \) |
| 47 | \( 1 + 1.07e10T^{2} \) |
| 53 | \( 1 + 2.21e10T^{2} \) |
| 59 | \( 1 - 4.21e10T^{2} \) |
| 61 | \( 1 + 4.20e5T + 5.15e10T^{2} \) |
| 67 | \( 1 - 1.72e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 1.28e11T^{2} \) |
| 73 | \( 1 + 6.38e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 - 2.04e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + 3.26e11T^{2} \) |
| 89 | \( 1 - 4.96e11T^{2} \) |
| 97 | \( 1 + 5.64e4iT - 8.32e11T^{2} \) |
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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
−13.61587216363107662399393547845, −12.11307510215001099397436042564, −10.71927500835144163900893791338, −9.857817741933418514232921276411, −8.918601559971055287399416618298, −7.61802200503554585405045218592, −5.61024738007369046445097591718, −4.47441192740753461411610003764, −3.38256365635904452590860672978, −0.56162062531571553231000336270,
1.13585494306597733807108741887, 2.99090689549102054241367668730, 4.98282050826968890496107907231, 6.18036236124703964308565672939, 7.74370167361227050436387196653, 8.724339128276756680587988049490, 9.773930297832765476522475427445, 11.58208819437770289268308782701, 12.37682138915530620255834464222, 13.45768987889500577519856042939