| L(s) = 1 | + 8.92·3-s + 11.8·5-s − 9.85·7-s + 52.7·9-s + 39.0·11-s − 91.5·13-s + 105.·15-s − 37.1·17-s − 46.4·19-s − 87.9·21-s + 120.·23-s + 15.5·25-s + 229.·27-s + 27.2·29-s + 81.1·31-s + 348.·33-s − 116.·35-s − 10.9·37-s − 817.·39-s − 205.·41-s + 115.·43-s + 624.·45-s + 312.·47-s − 245.·49-s − 331.·51-s − 90.9·53-s + 463.·55-s + ⋯ |
| L(s) = 1 | + 1.71·3-s + 1.06·5-s − 0.532·7-s + 1.95·9-s + 1.07·11-s − 1.95·13-s + 1.82·15-s − 0.529·17-s − 0.561·19-s − 0.914·21-s + 1.09·23-s + 0.124·25-s + 1.63·27-s + 0.174·29-s + 0.470·31-s + 1.84·33-s − 0.564·35-s − 0.0488·37-s − 3.35·39-s − 0.782·41-s + 0.409·43-s + 2.07·45-s + 0.970·47-s − 0.716·49-s − 0.910·51-s − 0.235·53-s + 1.13·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.025640640\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.025640640\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 8.92T + 27T^{2} \) |
| 5 | \( 1 - 11.8T + 125T^{2} \) |
| 7 | \( 1 + 9.85T + 343T^{2} \) |
| 11 | \( 1 - 39.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 91.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 37.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 46.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 120.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 27.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 81.1T + 2.97e4T^{2} \) |
| 37 | \( 1 + 10.9T + 5.06e4T^{2} \) |
| 41 | \( 1 + 205.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 115.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 312.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 90.9T + 1.48e5T^{2} \) |
| 59 | \( 1 + 550.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 630.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 661.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 494.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 566.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 49.4T + 4.93e5T^{2} \) |
| 83 | \( 1 - 564.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.08e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 464.T + 9.12e5T^{2} \) |
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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
−13.16154726850348948241182793717, −12.20685414735519986830343525635, −10.30198576382715341401256737536, −9.418230919452329862065684237713, −8.994498049698335360907103452765, −7.51943253153869667627807186037, −6.49360770145709687340431244092, −4.59221321474450316069971580288, −3.01160054344601032192191593912, −1.96205488991068877670545027130,
1.96205488991068877670545027130, 3.01160054344601032192191593912, 4.59221321474450316069971580288, 6.49360770145709687340431244092, 7.51943253153869667627807186037, 8.994498049698335360907103452765, 9.418230919452329862065684237713, 10.30198576382715341401256737536, 12.20685414735519986830343525635, 13.16154726850348948241182793717
