L(s) = 1 | + 1.35e3·3-s − 1.22e4·5-s − 4.52e5·7-s + 2.35e5·9-s − 7.28e6·11-s + 2.80e7·13-s − 1.65e7·15-s − 1.45e8·17-s − 1.08e8·19-s − 6.12e8·21-s − 1.59e8·23-s − 1.07e9·25-s − 1.83e9·27-s + 4.40e9·29-s − 5.53e9·31-s − 9.85e9·33-s + 5.53e9·35-s + 1.25e10·37-s + 3.79e10·39-s + 1.51e10·41-s + 3.15e10·43-s − 2.87e9·45-s − 3.50e10·47-s + 1.07e11·49-s − 1.96e11·51-s − 1.09e11·53-s + 8.90e10·55-s + ⋯ |
L(s) = 1 | + 1.07·3-s − 0.349·5-s − 1.45·7-s + 0.147·9-s − 1.24·11-s + 1.61·13-s − 0.374·15-s − 1.45·17-s − 0.528·19-s − 1.55·21-s − 0.225·23-s − 0.877·25-s − 0.913·27-s + 1.37·29-s − 1.12·31-s − 1.32·33-s + 0.508·35-s + 0.801·37-s + 1.72·39-s + 0.496·41-s + 0.759·43-s − 0.0515·45-s − 0.474·47-s + 1.11·49-s − 1.56·51-s − 0.681·53-s + 0.433·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(7)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{15}{2})\) |
|
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 | \( 1 - 1.35e3T + 1.59e6T^{2} \) |
| 5 | \( 1 + 1.22e4T + 1.22e9T^{2} \) |
| 7 | \( 1 + 4.52e5T + 9.68e10T^{2} \) |
| 11 | \( 1 + 7.28e6T + 3.45e13T^{2} \) |
| 13 | \( 1 - 2.80e7T + 3.02e14T^{2} \) |
| 17 | \( 1 + 1.45e8T + 9.90e15T^{2} \) |
| 19 | \( 1 + 1.08e8T + 4.20e16T^{2} \) |
| 23 | \( 1 + 1.59e8T + 5.04e17T^{2} \) |
| 29 | \( 1 - 4.40e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + 5.53e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 1.25e10T + 2.43e20T^{2} \) |
| 41 | \( 1 - 1.51e10T + 9.25e20T^{2} \) |
| 43 | \( 1 - 3.15e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + 3.50e10T + 5.46e21T^{2} \) |
| 53 | \( 1 + 1.09e11T + 2.60e22T^{2} \) |
| 59 | \( 1 - 1.18e11T + 1.04e23T^{2} \) |
| 61 | \( 1 + 2.98e11T + 1.61e23T^{2} \) |
| 67 | \( 1 + 7.44e11T + 5.48e23T^{2} \) |
| 71 | \( 1 - 1.15e12T + 1.16e24T^{2} \) |
| 73 | \( 1 - 1.84e12T + 1.67e24T^{2} \) |
| 79 | \( 1 - 2.20e12T + 4.66e24T^{2} \) |
| 83 | \( 1 - 1.92e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + 5.50e12T + 2.19e25T^{2} \) |
| 97 | \( 1 - 1.35e13T + 6.73e25T^{2} \) |
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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
−15.45320566557911739585848381208, −13.66595312993762257271194376020, −12.93589264550886955375061171692, −10.84838437184010403092606988067, −9.240502414505030205043998468535, −8.090087681447768409918649465184, −6.28027051356634941902753545627, −3.77846720030831692429756819543, −2.54482209674031040691983170860, 0,
2.54482209674031040691983170860, 3.77846720030831692429756819543, 6.28027051356634941902753545627, 8.090087681447768409918649465184, 9.240502414505030205043998468535, 10.84838437184010403092606988067, 12.93589264550886955375061171692, 13.66595312993762257271194376020, 15.45320566557911739585848381208