L(s) = 1 | + (1 + 1.73i)5-s + (0.5 − 0.866i)7-s + (−1 + 1.73i)11-s + (−0.5 − 0.866i)13-s − 6·17-s + 5·19-s + (3 + 5.19i)23-s + (0.500 − 0.866i)25-s + (4 − 6.92i)29-s + (4 + 6.92i)31-s + 1.99·35-s − 5·37-s + (4 + 6.92i)41-s + (−2 + 3.46i)43-s + (−5 + 8.66i)47-s + ⋯ |
L(s) = 1 | + (0.447 + 0.774i)5-s + (0.188 − 0.327i)7-s + (−0.301 + 0.522i)11-s + (−0.138 − 0.240i)13-s − 1.45·17-s + 1.14·19-s + (0.625 + 1.08i)23-s + (0.100 − 0.173i)25-s + (0.742 − 1.28i)29-s + (0.718 + 1.24i)31-s + 0.338·35-s − 0.821·37-s + (0.624 + 1.08i)41-s + (−0.304 + 0.528i)43-s + (−0.729 + 1.26i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.736555151\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.736555151\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-0.5 + 0.866i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4 + 6.92i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-4 - 6.92i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 5T + 37T^{2} \) |
| 41 | \( 1 + (-4 - 6.92i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2 - 3.46i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (5 - 8.66i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 4T + 53T^{2} \) |
| 59 | \( 1 + (-7 - 12.1i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.5 - 2.59i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.5 + 11.2i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 - 9T + 73T^{2} \) |
| 79 | \( 1 + (-5.5 + 9.52i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (6 - 10.3i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.213002221579420823555174865191, −8.165086835023029320944794891760, −7.47353419591047816088231299053, −6.74997269872407764580730962900, −6.13195104448681677519464048472, −5.03824587906532828814171391247, −4.42350631041778092247109224710, −3.14992470577573226353437722162, −2.50717646661653915329231785718, −1.26214441446488709386656294056,
0.60258635956818946545576541919, 1.88150168958661787213519950379, 2.81009910544668452973445177859, 3.96671119725420536024512769369, 5.08740616703089938654967419247, 5.27194394988949148319469265090, 6.48146365822929023586585994050, 7.05160048666413324128826862176, 8.219724971534786197642094283478, 8.749620002457753040744826696389