L(s) = 1 | + (−4.24 − 3i)3-s + 5.65·5-s + 16.9i·7-s + (8.99 + 25.4i)9-s + 30i·11-s − 72i·13-s + (−24 − 16.9i)15-s − 50.9i·17-s − 25.4·19-s + (50.9 − 71.9i)21-s − 144·23-s − 93·25-s + (38.1 − 134. i)27-s + 5.65·29-s + 220. i·31-s + ⋯ |
L(s) = 1 | + (−0.816 − 0.577i)3-s + 0.505·5-s + 0.916i·7-s + (0.333 + 0.942i)9-s + 0.822i·11-s − 1.53i·13-s + (−0.413 − 0.292i)15-s − 0.726i·17-s − 0.307·19-s + (0.529 − 0.748i)21-s − 1.30·23-s − 0.743·25-s + (0.272 − 0.962i)27-s + 0.0362·29-s + 1.27i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.3078461414\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3078461414\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (4.24 + 3i)T \) |
good | 5 | \( 1 - 5.65T + 125T^{2} \) |
| 7 | \( 1 - 16.9iT - 343T^{2} \) |
| 11 | \( 1 - 30iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 72iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 50.9iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 25.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 144T + 1.21e4T^{2} \) |
| 29 | \( 1 - 5.65T + 2.43e4T^{2} \) |
| 31 | \( 1 - 220. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 72iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 305. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 229.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 576T + 1.03e5T^{2} \) |
| 53 | \( 1 + 514.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 414iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 504iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 789.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 720T + 3.57e5T^{2} \) |
| 73 | \( 1 + 178T + 3.89e5T^{2} \) |
| 79 | \( 1 - 967. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 438iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 865. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 650T + 9.12e5T^{2} \) |
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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
−11.42781723261180344783342380564, −10.31599360061126176686639583731, −9.702187823605219810596129867316, −8.343218639126631751172597323835, −7.51505926995414771743941616050, −6.32584564579157437845224434222, −5.61602339527864029087998131985, −4.74086984240893309868163592310, −2.78973873002070409550126386262, −1.61469915683566450054172956076,
0.11271055687836309254715393360, 1.73892460011998840207050988012, 3.77767869185827828662869807685, 4.43271045994656069647611883641, 5.89012125519471838152103540997, 6.39626650088952568682881263892, 7.63657467797307740312754406415, 8.955929288825778851079014502552, 9.799179679948583256411289085764, 10.56484910473972042310922415833