| L(s) = 1 | − 19.9·3-s − 106.·5-s − 49·7-s + 156.·9-s + 452.·11-s − 886.·13-s + 2.12e3·15-s + 297.·17-s − 2.28e3·19-s + 978.·21-s + 555.·23-s + 8.14e3·25-s + 1.73e3·27-s + 8.26e3·29-s + 4.24e3·31-s − 9.03e3·33-s + 5.20e3·35-s + 758.·37-s + 1.77e4·39-s + 1.72e4·41-s + 5.37e3·43-s − 1.65e4·45-s − 2.56e4·47-s + 2.40e3·49-s − 5.95e3·51-s − 1.08e4·53-s − 4.80e4·55-s + ⋯ |
| L(s) = 1 | − 1.28·3-s − 1.89·5-s − 0.377·7-s + 0.642·9-s + 1.12·11-s − 1.45·13-s + 2.43·15-s + 0.250·17-s − 1.45·19-s + 0.484·21-s + 0.218·23-s + 2.60·25-s + 0.458·27-s + 1.82·29-s + 0.793·31-s − 1.44·33-s + 0.717·35-s + 0.0911·37-s + 1.86·39-s + 1.59·41-s + 0.443·43-s − 1.22·45-s − 1.69·47-s + 0.142·49-s − 0.320·51-s − 0.529·53-s − 2.14·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 49T \) |
| good | 3 | \( 1 + 19.9T + 243T^{2} \) |
| 5 | \( 1 + 106.T + 3.12e3T^{2} \) |
| 11 | \( 1 - 452.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 886.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 297.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.28e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 555.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 8.26e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.24e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 758.T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.72e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 5.37e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.56e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.08e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.79e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 8.46e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.43e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.11e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.00e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.53e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.43e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.03e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.30e4T + 8.58e9T^{2} \) |
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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
−10.11721176077500943723318057738, −8.843546879624781213114835570731, −7.87752944288951068854735279036, −6.89552665459904884929670863296, −6.28703432154717144897977359359, −4.72764644962973556402963702769, −4.30811122420106708279243856438, −2.95150830878000964017584691216, −0.827392042152277703753671441041, 0,
0.827392042152277703753671441041, 2.95150830878000964017584691216, 4.30811122420106708279243856438, 4.72764644962973556402963702769, 6.28703432154717144897977359359, 6.89552665459904884929670863296, 7.87752944288951068854735279036, 8.843546879624781213114835570731, 10.11721176077500943723318057738
