L(s) = 1 | + (−0.456 − 2.96i)3-s + (4.61 − 7.99i)5-s + (−5.33 + 3.07i)7-s + (−8.58 + 2.70i)9-s + (−3.70 + 2.13i)11-s + (0.869 − 1.50i)13-s + (−25.8 − 10.0i)15-s + 12.3·17-s − 33.9i·19-s + (11.5 + 14.4i)21-s + (−3.35 − 1.93i)23-s + (−30.1 − 52.1i)25-s + (11.9 + 24.2i)27-s + (17.8 + 30.9i)29-s + (38.8 + 22.4i)31-s + ⋯ |
L(s) = 1 | + (−0.152 − 0.988i)3-s + (0.923 − 1.59i)5-s + (−0.761 + 0.439i)7-s + (−0.953 + 0.300i)9-s + (−0.336 + 0.194i)11-s + (0.0668 − 0.115i)13-s + (−1.72 − 0.669i)15-s + 0.726·17-s − 1.78i·19-s + (0.550 + 0.685i)21-s + (−0.145 − 0.0841i)23-s + (−1.20 − 2.08i)25-s + (0.442 + 0.896i)27-s + (0.615 + 1.06i)29-s + (1.25 + 0.723i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.537 + 0.843i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.537 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.628693 - 1.14591i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.628693 - 1.14591i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.456 + 2.96i)T \) |
good | 5 | \( 1 + (-4.61 + 7.99i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (5.33 - 3.07i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (3.70 - 2.13i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-0.869 + 1.50i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 12.3T + 289T^{2} \) |
| 19 | \( 1 + 33.9iT - 361T^{2} \) |
| 23 | \( 1 + (3.35 + 1.93i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-17.8 - 30.9i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-38.8 - 22.4i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 32.7T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-21.8 + 37.8i)T + (-840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-33.9 + 19.5i)T + (924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-39.8 + 23.0i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 - 46.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-23.2 - 13.4i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-23.4 - 40.6i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (56.9 + 32.9i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 96.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 14.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-34.3 + 19.8i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-81.7 + 47.1i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 81.8T + 7.92e3T^{2} \) |
| 97 | \( 1 + (7.99 + 13.8i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.55547153832009357504493785514, −12.05232878115219662778458841360, −10.37676218749022329835850761122, −9.114016008399341321143172426941, −8.552677892063699416170965133692, −7.03357197289673777925829692219, −5.82530849102904714351119061198, −4.99350331169179216802146706196, −2.54212287683794237982540762268, −0.898585450528155957819710722484,
2.76542418881860674210321878122, 3.82408886824285930302862034991, 5.77055316125661783569707657419, 6.41081856232672334286942740709, 7.893363551150228800884641510526, 9.686404584841958779243584018201, 10.08740699254254703480522363217, 10.79583110136142655983404326054, 11.98841326221543560365604245677, 13.56839672280234191243500143409