L(s) = 1 | + (−1.90 + 0.613i)2-s + (3.24 − 2.33i)4-s − 6.64i·5-s − 0.470i·7-s + (−4.74 + 6.43i)8-s + (4.07 + 12.6i)10-s − 10.1·11-s + 3.45i·13-s + (0.288 + 0.896i)14-s + (5.09 − 15.1i)16-s − 14.2i·17-s + (−13.6 + 13.2i)19-s + (−15.5 − 21.5i)20-s + (19.3 − 6.24i)22-s − 25.5·23-s + ⋯ |
L(s) = 1 | + (−0.951 + 0.306i)2-s + (0.811 − 0.583i)4-s − 1.32i·5-s − 0.0672i·7-s + (−0.593 + 0.804i)8-s + (0.407 + 1.26i)10-s − 0.924·11-s + 0.265i·13-s + (0.0206 + 0.0640i)14-s + (0.318 − 0.948i)16-s − 0.839i·17-s + (−0.717 + 0.696i)19-s + (−0.775 − 1.07i)20-s + (0.880 − 0.283i)22-s − 1.11·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.702 - 0.711i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.702 - 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1213039496\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1213039496\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.90 - 0.613i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (13.6 - 13.2i)T \) |
good | 5 | \( 1 + 6.64iT - 25T^{2} \) |
| 7 | \( 1 + 0.470iT - 49T^{2} \) |
| 11 | \( 1 + 10.1T + 121T^{2} \) |
| 13 | \( 1 - 3.45iT - 169T^{2} \) |
| 17 | \( 1 + 14.2iT - 289T^{2} \) |
| 23 | \( 1 + 25.5T + 529T^{2} \) |
| 29 | \( 1 + 14.1T + 841T^{2} \) |
| 31 | \( 1 - 19.3T + 961T^{2} \) |
| 37 | \( 1 - 30.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 25.6T + 1.68e3T^{2} \) |
| 43 | \( 1 - 38.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 16.5T + 2.20e3T^{2} \) |
| 53 | \( 1 + 41.6T + 2.80e3T^{2} \) |
| 59 | \( 1 - 31.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 3.53T + 3.72e3T^{2} \) |
| 67 | \( 1 - 13.9T + 4.48e3T^{2} \) |
| 71 | \( 1 - 128. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 71.6T + 5.32e3T^{2} \) |
| 79 | \( 1 + 100.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 58.6T + 6.88e3T^{2} \) |
| 89 | \( 1 + 108.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 124. iT - 9.40e3T^{2} \) |
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
−10.29298112665637107804935570114, −9.671241365581507921254668518967, −8.764561454276660643497551293146, −8.158107367708437671736154992626, −7.39895726496422776519881342008, −6.15789107224442480401920820790, −5.33384563974289574062864040806, −4.34375471168127229665592324417, −2.54082880135795141338590992974, −1.25910362799159007493574612098,
0.06028258608397030125324523799, 2.08701194120030859651984597191, 2.88529688228188133419374331427, 4.00573745881317192903640879798, 5.78338848230684648982574992832, 6.60746974838572191247371550257, 7.48924964280374038195355393083, 8.158169815538998812058828759547, 9.163290697868653793380891166668, 10.26946248288066031916871881736