L(s) = 1 | + (−2.64 + 4.47i)3-s − 8·4-s − 11.1i·5-s + 18.5·7-s + (−13.0 − 23.6i)9-s + 11.8i·11-s + (21.1 − 35.7i)12-s + 84.6·13-s + (50.0 + 29.5i)15-s + 64·16-s − 102. i·17-s + 89.4i·20-s + (−49.0 + 82.8i)21-s − 125.·25-s + (140. + 4.47i)27-s − 148.·28-s + ⋯ |
L(s) = 1 | + (−0.509 + 0.860i)3-s − 4-s − 0.999i·5-s + 0.999·7-s + (−0.481 − 0.876i)9-s + 0.324i·11-s + (0.509 − 0.860i)12-s + 1.80·13-s + (0.860 + 0.509i)15-s + 16-s − 1.46i·17-s + 0.999i·20-s + (−0.509 + 0.860i)21-s − 1.00·25-s + (0.999 + 0.0318i)27-s − 0.999·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 + 0.509i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.860 + 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.08372 - 0.296564i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.08372 - 0.296564i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (2.64 - 4.47i)T \) |
| 5 | \( 1 + 11.1iT \) |
| 7 | \( 1 - 18.5T \) |
good | 2 | \( 1 + 8T^{2} \) |
| 11 | \( 1 - 11.8iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 84.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 102. iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 6.85e3T^{2} \) |
| 23 | \( 1 + 1.21e4T^{2} \) |
| 29 | \( 1 + 307. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 2.97e4T^{2} \) |
| 37 | \( 1 - 5.06e4T^{2} \) |
| 41 | \( 1 + 6.89e4T^{2} \) |
| 43 | \( 1 - 7.95e4T^{2} \) |
| 47 | \( 1 - 178. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 1.48e5T^{2} \) |
| 59 | \( 1 + 2.05e5T^{2} \) |
| 61 | \( 1 - 2.26e5T^{2} \) |
| 67 | \( 1 - 3.00e5T^{2} \) |
| 71 | \( 1 - 863. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 1.13e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 236T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.51e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 7.04e5T^{2} \) |
| 97 | \( 1 + 963.T + 9.12e5T^{2} \) |
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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
−13.33145176651158429888769440093, −11.99631013762628393303348693809, −11.15539286371327506422488200490, −9.792223646675403975162747027153, −8.947711014294686640449541922188, −8.083948643330592681195056613167, −5.81046875853650777734471614348, −4.83257541359668209823974145049, −4.00620415773054123544866592204, −0.824560590213629432363638603045,
1.42538772465813365357571217180, 3.70366538254294274806929077678, 5.43131296013000440657097460426, 6.48505104499050042421961108682, 7.963437349696721112276010922968, 8.675200589701545521645104493429, 10.63846150245199012821570316289, 11.05849793207044871144049849024, 12.42812687668531733905473248784, 13.50117854374726516515483662242