L(s) = 1 | + (−138. + 397. i)3-s + 8.83e3·5-s + (1.66e4 + 4.12e4i)7-s + (−1.38e5 − 1.09e5i)9-s − 6.74e5i·11-s + 1.36e6i·13-s + (−1.22e6 + 3.51e6i)15-s − 7.92e6·17-s + 8.91e6i·19-s + (−1.86e7 + 9.31e5i)21-s − 2.25e7i·23-s + 2.91e7·25-s + (6.28e7 − 4.00e7i)27-s + 8.14e7i·29-s + 2.35e8i·31-s + ⋯ |
L(s) = 1 | + (−0.328 + 0.944i)3-s + 1.26·5-s + (0.374 + 0.927i)7-s + (−0.784 − 0.620i)9-s − 1.26i·11-s + 1.01i·13-s + (−0.414 + 1.19i)15-s − 1.35·17-s + 0.826i·19-s + (−0.998 + 0.0497i)21-s − 0.730i·23-s + 0.597·25-s + (0.843 − 0.537i)27-s + 0.737i·29-s + 1.47i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0497i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0497i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(\approx\) |
\(0.0303002 - 1.21636i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0303002 - 1.21636i\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (138. - 397. i)T \) |
| 7 | \( 1 + (-1.66e4 - 4.12e4i)T \) |
good | 5 | \( 1 - 8.83e3T + 4.88e7T^{2} \) |
| 11 | \( 1 + 6.74e5iT - 2.85e11T^{2} \) |
| 13 | \( 1 - 1.36e6iT - 1.79e12T^{2} \) |
| 17 | \( 1 + 7.92e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 8.91e6iT - 1.16e14T^{2} \) |
| 23 | \( 1 + 2.25e7iT - 9.52e14T^{2} \) |
| 29 | \( 1 - 8.14e7iT - 1.22e16T^{2} \) |
| 31 | \( 1 - 2.35e8iT - 2.54e16T^{2} \) |
| 37 | \( 1 + 5.33e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 9.52e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.43e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + 2.52e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 2.77e9iT - 9.26e18T^{2} \) |
| 59 | \( 1 + 6.78e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 4.33e8iT - 4.35e19T^{2} \) |
| 67 | \( 1 + 1.56e10T + 1.22e20T^{2} \) |
| 71 | \( 1 - 6.69e9iT - 2.31e20T^{2} \) |
| 73 | \( 1 + 2.11e10iT - 3.13e20T^{2} \) |
| 79 | \( 1 + 2.95e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 5.39e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 3.45e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 2.62e10iT - 7.15e21T^{2} \) |
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
−12.44416681597809760619148673416, −11.25933545771364164985984947449, −10.46356464115410384432020901299, −9.108208164991186317442896220014, −8.752827729856977927911754270534, −6.38685238043448935082433033248, −5.69224457139641425631065432621, −4.55945671446792886668677945677, −2.93343049154708524572066440663, −1.65064364096366832760526856247,
0.28338965996771373404204211722, 1.57831291114832993941573090927, 2.44603071128942118258783354325, 4.59007945012489342795290009722, 5.79713228194128137917008970991, 6.90426409293668671589127672195, 7.80137258489394123627798155038, 9.367354732760526444099794338089, 10.44730211718982948553228439480, 11.43826486993841956198588898082