L(s) = 1 | + (2.15 + 0.632i)2-s + (0.142 − 0.989i)3-s + (2.55 + 1.64i)4-s + (−0.146 − 0.319i)5-s + (0.932 − 2.04i)6-s + (−0.489 − 0.143i)7-s + (1.52 + 1.76i)8-s + (−0.959 − 0.281i)9-s + (−0.112 − 0.781i)10-s + (0.961 + 2.10i)11-s + (1.99 − 2.29i)12-s + (−1.11 + 1.28i)13-s + (−0.964 − 0.619i)14-s + (−0.337 + 0.0990i)15-s + (−0.348 − 0.763i)16-s + (−3.94 + 2.53i)17-s + ⋯ |
L(s) = 1 | + (1.52 + 0.447i)2-s + (0.0821 − 0.571i)3-s + (1.27 + 0.821i)4-s + (−0.0653 − 0.143i)5-s + (0.380 − 0.833i)6-s + (−0.185 − 0.0543i)7-s + (0.540 + 0.623i)8-s + (−0.319 − 0.0939i)9-s + (−0.0355 − 0.247i)10-s + (0.289 + 0.634i)11-s + (0.574 − 0.663i)12-s + (−0.309 + 0.357i)13-s + (−0.257 − 0.165i)14-s + (−0.0871 + 0.0255i)15-s + (−0.0872 − 0.190i)16-s + (−0.956 + 0.614i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.133i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.133i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.43233 + 0.163098i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.43233 + 0.163098i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.142 + 0.989i)T \) |
| 67 | \( 1 + (-8.16 + 0.560i)T \) |
good | 2 | \( 1 + (-2.15 - 0.632i)T + (1.68 + 1.08i)T^{2} \) |
| 5 | \( 1 + (0.146 + 0.319i)T + (-3.27 + 3.77i)T^{2} \) |
| 7 | \( 1 + (0.489 + 0.143i)T + (5.88 + 3.78i)T^{2} \) |
| 11 | \( 1 + (-0.961 - 2.10i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (1.11 - 1.28i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (3.94 - 2.53i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (2.21 - 0.651i)T + (15.9 - 10.2i)T^{2} \) |
| 23 | \( 1 + (-0.229 + 1.59i)T + (-22.0 - 6.47i)T^{2} \) |
| 29 | \( 1 + 2.16T + 29T^{2} \) |
| 31 | \( 1 + (-1.40 - 1.62i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 - 5.02T + 37T^{2} \) |
| 41 | \( 1 + (3.00 - 1.92i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (-5.14 + 3.30i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + (0.984 - 6.84i)T + (-45.0 - 13.2i)T^{2} \) |
| 53 | \( 1 + (0.767 + 0.493i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-3.80 - 4.38i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-3.68 + 8.06i)T + (-39.9 - 46.1i)T^{2} \) |
| 71 | \( 1 + (-7.41 - 4.76i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-6.17 + 13.5i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (0.963 - 1.11i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (1.73 + 3.80i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (-1.97 - 13.7i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + 5.56T + 97T^{2} \) |
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
−12.68310155110996931069210766148, −12.04730559609790341975040512867, −10.92334839194692423695797860697, −9.423659700187687481094855389837, −8.119078654297551339110505994203, −6.84663888393437562152620937464, −6.33070252182462706904511806556, −4.91694772098090392672248154346, −3.95921136369444638555479992168, −2.36987034381965948139123368446,
2.58882953805122234302431307310, 3.67576006849097792132415020431, 4.75227698074216987659647061661, 5.76243498973167149822851529751, 6.89043361091053579567435029753, 8.551711253927755841186877445878, 9.693358439711783898793545449956, 11.02984475340431306799019346371, 11.38569941320319984236249661559, 12.61953731819299018491116607637