| L(s) = 1 | − 3·3-s − 10.1·5-s + 9·9-s − 37.8·11-s + 74.0·13-s + 30.4·15-s + 44.6·17-s − 139.·19-s + 57.8·23-s − 21.8·25-s − 27·27-s + 39.5·29-s + 254.·31-s + 113.·33-s + 18.1·37-s − 222.·39-s + 420.·41-s − 382.·43-s − 91.4·45-s + 484.·47-s − 134.·51-s + 36.0·53-s + 383.·55-s + 418.·57-s + 145.·59-s − 78.1·61-s − 752.·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.908·5-s + 0.333·9-s − 1.03·11-s + 1.58·13-s + 0.524·15-s + 0.637·17-s − 1.68·19-s + 0.524·23-s − 0.174·25-s − 0.192·27-s + 0.252·29-s + 1.47·31-s + 0.598·33-s + 0.0804·37-s − 0.912·39-s + 1.60·41-s − 1.35·43-s − 0.302·45-s + 1.50·47-s − 0.367·51-s + 0.0935·53-s + 0.941·55-s + 0.971·57-s + 0.321·59-s − 0.164·61-s − 1.43·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 10.1T + 125T^{2} \) |
| 11 | \( 1 + 37.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 74.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 44.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 139.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 57.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 39.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 254.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 18.1T + 5.06e4T^{2} \) |
| 41 | \( 1 - 420.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 382.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 484.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 36.0T + 1.48e5T^{2} \) |
| 59 | \( 1 - 145.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 78.1T + 2.26e5T^{2} \) |
| 67 | \( 1 + 273.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 717.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.10e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.08e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 404.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.35e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 681.T + 9.12e5T^{2} \) |
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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
−8.721791548321619366463841877122, −8.215107844107113752991814230694, −7.39258881800031844839904949108, −6.37006264386044612882830252111, −5.69671048687724483648362094583, −4.55052088815090697173473280582, −3.85841823848544116395014416687, −2.68419286622225052212091786806, −1.13788046808440986753111833155, 0,
1.13788046808440986753111833155, 2.68419286622225052212091786806, 3.85841823848544116395014416687, 4.55052088815090697173473280582, 5.69671048687724483648362094583, 6.37006264386044612882830252111, 7.39258881800031844839904949108, 8.215107844107113752991814230694, 8.721791548321619366463841877122
