| 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
−13.50117854374726516515483662242, −12.42812687668531733905473248784, −11.05849793207044871144049849024, −10.63846150245199012821570316289, −8.675200589701545521645104493429, −7.963437349696721112276010922968, −6.48505104499050042421961108682, −5.43131296013000440657097460426, −3.70366538254294274806929077678, −1.42538772465813365357571217180,
0.824560590213629432363638603045, 4.00620415773054123544866592204, 4.83257541359668209823974145049, 5.81046875853650777734471614348, 8.083948643330592681195056613167, 8.947711014294686640449541922188, 9.792223646675403975162747027153, 11.15539286371327506422488200490, 11.99631013762628393303348693809, 13.33145176651158429888769440093
