| L(s) = 1 | + 6.31·5-s − 29.5·7-s + 49.1·11-s + 18.2·13-s − 63.7·17-s − 52.0·19-s + 149.·23-s − 85.0·25-s − 119.·29-s + 303.·31-s − 186.·35-s + 170.·37-s − 43.6·41-s + 446.·43-s + 221.·47-s + 531.·49-s + 67.3·53-s + 310.·55-s + 467.·59-s − 475.·61-s + 115.·65-s − 240.·67-s − 550.·71-s + 552.·73-s − 1.45e3·77-s + 913.·79-s − 667.·83-s + ⋯ |
| L(s) = 1 | + 0.565·5-s − 1.59·7-s + 1.34·11-s + 0.389·13-s − 0.910·17-s − 0.628·19-s + 1.35·23-s − 0.680·25-s − 0.764·29-s + 1.75·31-s − 0.902·35-s + 0.758·37-s − 0.166·41-s + 1.58·43-s + 0.688·47-s + 1.55·49-s + 0.174·53-s + 0.761·55-s + 1.03·59-s − 0.997·61-s + 0.220·65-s − 0.439·67-s − 0.920·71-s + 0.886·73-s − 2.15·77-s + 1.30·79-s − 0.883·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.888334654\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.888334654\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 6.31T + 125T^{2} \) |
| 7 | \( 1 + 29.5T + 343T^{2} \) |
| 11 | \( 1 - 49.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 18.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 63.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 52.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 149.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 119.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 303.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 170.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 43.6T + 6.89e4T^{2} \) |
| 43 | \( 1 - 446.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 221.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 67.3T + 1.48e5T^{2} \) |
| 59 | \( 1 - 467.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 475.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 240.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 550.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 552.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 913.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 667.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.43e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.66e3T + 9.12e5T^{2} \) |
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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
−10.00042439817061037683398437820, −9.252494916014545904017562998698, −8.804305126348422211541831978569, −7.27185969599602486383147260949, −6.35806543162409382689893686087, −6.05624632506419231947587701052, −4.44912437290931287329311567645, −3.49518654413670596889366316628, −2.34421694315093989926314800867, −0.807243550049729041496155745530,
0.807243550049729041496155745530, 2.34421694315093989926314800867, 3.49518654413670596889366316628, 4.44912437290931287329311567645, 6.05624632506419231947587701052, 6.35806543162409382689893686087, 7.27185969599602486383147260949, 8.804305126348422211541831978569, 9.252494916014545904017562998698, 10.00042439817061037683398437820
