L(s) = 1 | + (2.82 + i)3-s − 7i·7-s + (7.00 + 5.65i)9-s + 8.48i·11-s − 25i·13-s + 25.4·17-s − 7·19-s + (7 − 19.7i)21-s − 25.4·23-s + (14.1 + 23.0i)27-s − 42.4i·29-s + 7·31-s + (−8.48 + 24i)33-s − 2i·37-s + (25 − 70.7i)39-s + ⋯ |
L(s) = 1 | + (0.942 + 0.333i)3-s − i·7-s + (0.777 + 0.628i)9-s + 0.771i·11-s − 1.92i·13-s + 1.49·17-s − 0.368·19-s + (0.333 − 0.942i)21-s − 1.10·23-s + (0.523 + 0.851i)27-s − 1.46i·29-s + 0.225·31-s + (−0.257 + 0.727i)33-s − 0.0540i·37-s + (0.641 − 1.81i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.694 + 0.719i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.694 + 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.752909990\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.752909990\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.82 - i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 7iT - 49T^{2} \) |
| 11 | \( 1 - 8.48iT - 121T^{2} \) |
| 13 | \( 1 + 25iT - 169T^{2} \) |
| 17 | \( 1 - 25.4T + 289T^{2} \) |
| 19 | \( 1 + 7T + 361T^{2} \) |
| 23 | \( 1 + 25.4T + 529T^{2} \) |
| 29 | \( 1 + 42.4iT - 841T^{2} \) |
| 31 | \( 1 - 7T + 961T^{2} \) |
| 37 | \( 1 + 2iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 8.48iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 41iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 2.20e3T^{2} \) |
| 53 | \( 1 - 59.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + 33.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + T + 3.72e3T^{2} \) |
| 67 | \( 1 - 17iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 42.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 70iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 58T + 6.24e3T^{2} \) |
| 83 | \( 1 - 118.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 135. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 49iT - 9.40e3T^{2} \) |
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
−9.700919391542012521260181776610, −8.475707010246377208937597575725, −7.70092893531156189118766607370, −7.47612019808530877251120358993, −6.06019561065432534927676230370, −5.03854970225110971379856822461, −4.02391856487902037429145933374, −3.33440985171455197597220348942, −2.19666061526105055707763400553, −0.76144653249369402782094852072,
1.37836683145738001812477155565, 2.34940723502259828140793591411, 3.36327907802814100333814439448, 4.27732212041454734113724923231, 5.55600006775093966164655338703, 6.40411264189723278383452874385, 7.24786952364835124440880722676, 8.233742991471500324935315339049, 8.799878682627010753915350454446, 9.424166377990727530241271134791