L(s) = 1 | + 2.56·3-s + 5-s + 7-s + 3.56·9-s − 2.56·11-s + 4.56·13-s + 2.56·15-s − 4.56·17-s − 1.12·19-s + 2.56·21-s + 5.12·23-s + 25-s + 1.43·27-s − 5.68·29-s − 6.56·33-s + 35-s + 6·37-s + 11.6·39-s − 3.12·41-s − 9.12·43-s + 3.56·45-s − 3.68·47-s + 49-s − 11.6·51-s + 3.12·53-s − 2.56·55-s − 2.87·57-s + ⋯ |
L(s) = 1 | + 1.47·3-s + 0.447·5-s + 0.377·7-s + 1.18·9-s − 0.772·11-s + 1.26·13-s + 0.661·15-s − 1.10·17-s − 0.257·19-s + 0.558·21-s + 1.06·23-s + 0.200·25-s + 0.276·27-s − 1.05·29-s − 1.14·33-s + 0.169·35-s + 0.986·37-s + 1.87·39-s − 0.487·41-s − 1.39·43-s + 0.530·45-s − 0.537·47-s + 0.142·49-s − 1.63·51-s + 0.428·53-s − 0.345·55-s − 0.381·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.525502063\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.525502063\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
good | 3 | \( 1 - 2.56T + 3T^{2} \) |
| 11 | \( 1 + 2.56T + 11T^{2} \) |
| 13 | \( 1 - 4.56T + 13T^{2} \) |
| 17 | \( 1 + 4.56T + 17T^{2} \) |
| 19 | \( 1 + 1.12T + 19T^{2} \) |
| 23 | \( 1 - 5.12T + 23T^{2} \) |
| 29 | \( 1 + 5.68T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 3.12T + 41T^{2} \) |
| 43 | \( 1 + 9.12T + 43T^{2} \) |
| 47 | \( 1 + 3.68T + 47T^{2} \) |
| 53 | \( 1 - 3.12T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 9.36T + 61T^{2} \) |
| 67 | \( 1 - 6.24T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 4.24T + 73T^{2} \) |
| 79 | \( 1 - 6.56T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 - 7.12T + 89T^{2} \) |
| 97 | \( 1 + 14.8T + 97T^{2} \) |
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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
−10.71852278879301063064640206695, −9.665051256331339298262348336268, −8.827835147304446976532813852039, −8.351613035344276371663628470056, −7.40369311082293681903432731110, −6.31541053035408999850880288606, −5.04595920059395037042341789054, −3.84123208348491185714193855596, −2.78921232307861910493038679196, −1.74497648316645092426880643265,
1.74497648316645092426880643265, 2.78921232307861910493038679196, 3.84123208348491185714193855596, 5.04595920059395037042341789054, 6.31541053035408999850880288606, 7.40369311082293681903432731110, 8.351613035344276371663628470056, 8.827835147304446976532813852039, 9.665051256331339298262348336268, 10.71852278879301063064640206695