L(s) = 1 | − 30.3·3-s + 100.·5-s + 49·7-s + 676.·9-s − 140.·11-s − 725.·13-s − 3.04e3·15-s − 365.·17-s − 202.·19-s − 1.48e3·21-s − 2.73e3·23-s + 6.95e3·25-s − 1.31e4·27-s − 2.17e3·29-s + 8.95e3·31-s + 4.25e3·33-s + 4.91e3·35-s + 2.60e3·37-s + 2.19e4·39-s + 1.42e4·41-s − 1.12e4·43-s + 6.79e4·45-s − 1.77e4·47-s + 2.40e3·49-s + 1.10e4·51-s − 2.59e4·53-s − 1.40e4·55-s + ⋯ |
L(s) = 1 | − 1.94·3-s + 1.79·5-s + 0.377·7-s + 2.78·9-s − 0.349·11-s − 1.19·13-s − 3.49·15-s − 0.306·17-s − 0.128·19-s − 0.735·21-s − 1.07·23-s + 2.22·25-s − 3.47·27-s − 0.480·29-s + 1.67·31-s + 0.680·33-s + 0.678·35-s + 0.312·37-s + 2.31·39-s + 1.32·41-s − 0.926·43-s + 5.00·45-s − 1.17·47-s + 0.142·49-s + 0.597·51-s − 1.26·53-s − 0.628·55-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 | \( 1 + 30.3T + 243T^{2} \) |
| 5 | \( 1 - 100.T + 3.12e3T^{2} \) |
| 11 | \( 1 + 140.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 725.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 365.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 202.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.73e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.17e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.95e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.60e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.42e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.12e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.77e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.59e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.76e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.93e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 854.T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.91e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.88e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.55e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.76e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.14e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.96e4T + 8.58e9T^{2} \) |
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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
−10.02283747287152598422246325253, −9.479933687489423568659286925513, −7.74042923600509338196506565569, −6.55037878710757008689712311679, −6.06819879486627583231173485404, −5.15802343356460381251762193850, −4.60699112099414143449580125186, −2.29653575399948151668037461329, −1.32392972213627602346754895890, 0,
1.32392972213627602346754895890, 2.29653575399948151668037461329, 4.60699112099414143449580125186, 5.15802343356460381251762193850, 6.06819879486627583231173485404, 6.55037878710757008689712311679, 7.74042923600509338196506565569, 9.479933687489423568659286925513, 10.02283747287152598422246325253