L(s) = 1 | + 15.5i·3-s + 20·5-s + 529. i·7-s − 243·9-s − 435. i·11-s + 341.·13-s + 311. i·15-s + 7.68e3·17-s − 4.30e3i·19-s − 8.25e3·21-s − 3.17e3i·23-s − 1.52e4·25-s − 3.78e3i·27-s − 1.94e4·29-s − 1.52e4i·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 0.160·5-s + 1.54i·7-s − 0.333·9-s − 0.327i·11-s + 0.155·13-s + 0.0923i·15-s + 1.56·17-s − 0.626i·19-s − 0.891·21-s − 0.260i·23-s − 0.974·25-s − 0.192i·27-s − 0.795·29-s − 0.513i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.4745247520\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4745247520\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 15.5iT \) |
good | 5 | \( 1 - 20T + 1.56e4T^{2} \) |
| 7 | \( 1 - 529. iT - 1.17e5T^{2} \) |
| 11 | \( 1 + 435. iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 341.T + 4.82e6T^{2} \) |
| 17 | \( 1 - 7.68e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + 4.30e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 + 3.17e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 1.94e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + 1.52e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 6.19e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 3.37e4T + 4.75e9T^{2} \) |
| 43 | \( 1 - 9.93e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 - 1.77e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 2.24e5T + 2.21e10T^{2} \) |
| 59 | \( 1 + 1.99e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 4.56e4T + 5.15e10T^{2} \) |
| 67 | \( 1 + 4.96e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 4.52e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 3.94e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 5.71e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 - 3.24e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 7.58e5T + 4.96e11T^{2} \) |
| 97 | \( 1 - 2.50e4T + 8.32e11T^{2} \) |
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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
−9.322550888255314622618754362969, −8.444292635689566685650756943902, −7.66885969381807754609573941470, −6.22037973511594729895242222637, −5.65253236282025085043465111414, −4.86890577807709829379306453980, −3.55008650128751391938943071632, −2.73912962020670114910676725592, −1.63373687944184562036791185365, −0.089950355601119815122092482150,
1.05174078842176385310008089776, 1.77211672189695461379539933619, 3.31286786846642098069004317777, 4.01865744487114155387028547725, 5.29132945798812208399986519856, 6.16680140570973856315486226531, 7.41626990842515354606735648984, 7.46347811403662293026094272913, 8.628108484342114762272315812287, 9.864616870413459492685337487957