| L(s) = 1 | − 3·3-s + 6.30·5-s + 7·7-s + 9·9-s − 48.9·11-s + 2.60·13-s − 18.9·15-s + 136.·17-s − 45.2·19-s − 21·21-s − 38.1·23-s − 85.2·25-s − 27·27-s − 52.7·29-s − 14.7·31-s + 146.·33-s + 44.1·35-s − 333.·37-s − 7.82·39-s + 227.·41-s + 398.·43-s + 56.7·45-s − 184.·47-s + 49·49-s − 410.·51-s − 359.·53-s − 308.·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.563·5-s + 0.377·7-s + 0.333·9-s − 1.34·11-s + 0.0556·13-s − 0.325·15-s + 1.95·17-s − 0.545·19-s − 0.218·21-s − 0.345·23-s − 0.682·25-s − 0.192·27-s − 0.337·29-s − 0.0856·31-s + 0.774·33-s + 0.213·35-s − 1.48·37-s − 0.0321·39-s + 0.865·41-s + 1.41·43-s + 0.187·45-s − 0.572·47-s + 0.142·49-s − 1.12·51-s − 0.932·53-s − 0.755·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 7T \) |
| good | 5 | \( 1 - 6.30T + 125T^{2} \) |
| 11 | \( 1 + 48.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 2.60T + 2.19e3T^{2} \) |
| 17 | \( 1 - 136.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 45.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 38.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 52.7T + 2.43e4T^{2} \) |
| 31 | \( 1 + 14.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 333.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 227.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 398.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 184.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 359.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 99.9T + 2.05e5T^{2} \) |
| 61 | \( 1 - 674.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 376.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.18e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 735.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 836.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 293.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.29e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 201.T + 9.12e5T^{2} \) |
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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
−8.840960751031803822892154813561, −7.83948456397982464219365481451, −7.38221321660641608555739934424, −6.03831266585576194335398888208, −5.58935902445925575368396872861, −4.81755376541141603813152416760, −3.61237687761207064539602668607, −2.42319536327476058552345856761, −1.34352988145364680011807430776, 0,
1.34352988145364680011807430776, 2.42319536327476058552345856761, 3.61237687761207064539602668607, 4.81755376541141603813152416760, 5.58935902445925575368396872861, 6.03831266585576194335398888208, 7.38221321660641608555739934424, 7.83948456397982464219365481451, 8.840960751031803822892154813561
