| L(s) = 1 | − 3·3-s + 32.4·7-s + 9·9-s − 41.5·11-s − 19.3·13-s − 132.·17-s − 80.4·19-s − 97.2·21-s − 134.·23-s − 27·27-s + 217.·29-s − 129.·31-s + 124.·33-s − 239.·37-s + 58.1·39-s + 371.·41-s + 476.·43-s + 44.5·47-s + 708.·49-s + 396.·51-s + 14.7·53-s + 241.·57-s − 186.·59-s − 609.·61-s + 291.·63-s + 332.·67-s + 403.·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.75·7-s + 0.333·9-s − 1.13·11-s − 0.413·13-s − 1.88·17-s − 0.971·19-s − 1.01·21-s − 1.21·23-s − 0.192·27-s + 1.39·29-s − 0.749·31-s + 0.657·33-s − 1.06·37-s + 0.238·39-s + 1.41·41-s + 1.68·43-s + 0.138·47-s + 2.06·49-s + 1.08·51-s + 0.0381·53-s + 0.560·57-s − 0.411·59-s − 1.28·61-s + 0.583·63-s + 0.605·67-s + 0.703·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.400920548\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.400920548\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 32.4T + 343T^{2} \) |
| 11 | \( 1 + 41.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 19.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 132.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 80.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 134.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 217.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 129.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 239.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 371.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 476.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 44.5T + 1.03e5T^{2} \) |
| 53 | \( 1 - 14.7T + 1.48e5T^{2} \) |
| 59 | \( 1 + 186.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 609.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 332.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 386.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.13e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.31e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 430.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.52e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 462.T + 9.12e5T^{2} \) |
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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
−8.427695495516863872843104108278, −7.927167996602161624371164226501, −7.13989515245160510302911794860, −6.21183152242341777761257595081, −5.35947366485571361209531323335, −4.62662901820064737079016425254, −4.19306896242383609338665871933, −2.41110814441400476416317585169, −1.93588763957682950215495832094, −0.52317655941659549929866699716,
0.52317655941659549929866699716, 1.93588763957682950215495832094, 2.41110814441400476416317585169, 4.19306896242383609338665871933, 4.62662901820064737079016425254, 5.35947366485571361209531323335, 6.21183152242341777761257595081, 7.13989515245160510302911794860, 7.927167996602161624371164226501, 8.427695495516863872843104108278
