L(s) = 1 | − 1.81·2-s + 3-s + 1.28·4-s + 5-s − 1.81·6-s − 7-s + 1.28·8-s + 9-s − 1.81·10-s + 11-s + 1.28·12-s + 5.10·13-s + 1.81·14-s + 15-s − 4.91·16-s − 1.91·17-s − 1.81·18-s + 3.33·19-s + 1.28·20-s − 21-s − 1.81·22-s − 2.28·23-s + 1.28·24-s + 25-s − 9.25·26-s + 27-s − 1.28·28-s + ⋯ |
L(s) = 1 | − 1.28·2-s + 0.577·3-s + 0.644·4-s + 0.447·5-s − 0.740·6-s − 0.377·7-s + 0.455·8-s + 0.333·9-s − 0.573·10-s + 0.301·11-s + 0.372·12-s + 1.41·13-s + 0.484·14-s + 0.258·15-s − 1.22·16-s − 0.464·17-s − 0.427·18-s + 0.765·19-s + 0.288·20-s − 0.218·21-s − 0.386·22-s − 0.477·23-s + 0.263·24-s + 0.200·25-s − 1.81·26-s + 0.192·27-s − 0.243·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.140024384\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.140024384\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 + 1.81T + 2T^{2} \) |
| 13 | \( 1 - 5.10T + 13T^{2} \) |
| 17 | \( 1 + 1.91T + 17T^{2} \) |
| 19 | \( 1 - 3.33T + 19T^{2} \) |
| 23 | \( 1 + 2.28T + 23T^{2} \) |
| 29 | \( 1 - 1.18T + 29T^{2} \) |
| 31 | \( 1 + 2.52T + 31T^{2} \) |
| 37 | \( 1 - 3.94T + 37T^{2} \) |
| 41 | \( 1 + 1.68T + 41T^{2} \) |
| 43 | \( 1 - 7.01T + 43T^{2} \) |
| 47 | \( 1 - 2.52T + 47T^{2} \) |
| 53 | \( 1 - 0.289T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 - 6.49T + 61T^{2} \) |
| 67 | \( 1 + 0.578T + 67T^{2} \) |
| 71 | \( 1 + 13.3T + 71T^{2} \) |
| 73 | \( 1 - 0.372T + 73T^{2} \) |
| 79 | \( 1 - 15.4T + 79T^{2} \) |
| 83 | \( 1 - 15.7T + 83T^{2} \) |
| 89 | \( 1 + 0.0297T + 89T^{2} \) |
| 97 | \( 1 - 15.8T + 97T^{2} \) |
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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
−9.443107139832556812593821332212, −9.145930444693450044123646600964, −8.349445127491029847310309488632, −7.60713544285575183569769036019, −6.69302223569455262545363937945, −5.85145447111588207445494802582, −4.45105020689498713526789056964, −3.40965620646151300683365652535, −2.10002421531779338431598503708, −1.00903709394830321841901430407,
1.00903709394830321841901430407, 2.10002421531779338431598503708, 3.40965620646151300683365652535, 4.45105020689498713526789056964, 5.85145447111588207445494802582, 6.69302223569455262545363937945, 7.60713544285575183569769036019, 8.349445127491029847310309488632, 9.145930444693450044123646600964, 9.443107139832556812593821332212