| L(s) = 1 | − 3.27e4·2-s − 2.18e7·3-s + 1.07e9·4-s − 1.27e11·5-s + 7.14e11·6-s − 1.05e13·7-s − 3.51e13·8-s − 1.42e14·9-s + 4.17e15·10-s + 6.26e15·11-s − 2.34e16·12-s + 2.37e17·13-s + 3.47e17·14-s + 2.77e18·15-s + 1.15e18·16-s + 6.39e17·17-s + 4.65e18·18-s − 7.77e19·19-s − 1.36e20·20-s + 2.30e20·21-s − 2.05e20·22-s − 1.47e21·23-s + 7.67e20·24-s + 1.15e22·25-s − 7.79e21·26-s + 1.65e22·27-s − 1.13e22·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.877·3-s + 0.5·4-s − 1.86·5-s + 0.620·6-s − 0.843·7-s − 0.353·8-s − 0.229·9-s + 1.32·10-s + 0.452·11-s − 0.438·12-s + 1.28·13-s + 0.596·14-s + 1.63·15-s + 0.250·16-s + 0.0541·17-s + 0.162·18-s − 1.17·19-s − 0.933·20-s + 0.739·21-s − 0.319·22-s − 1.15·23-s + 0.310·24-s + 2.48·25-s − 0.911·26-s + 1.07·27-s − 0.421·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(32-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+31/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(16)\) |
\(\approx\) |
\(0.3150115095\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3150115095\) |
| \(L(\frac{33}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 3.27e4T \) |
| good | 3 | \( 1 + 2.18e7T + 6.17e14T^{2} \) |
| 5 | \( 1 + 1.27e11T + 4.65e21T^{2} \) |
| 7 | \( 1 + 1.05e13T + 1.57e26T^{2} \) |
| 11 | \( 1 - 6.26e15T + 1.91e32T^{2} \) |
| 13 | \( 1 - 2.37e17T + 3.40e34T^{2} \) |
| 17 | \( 1 - 6.39e17T + 1.39e38T^{2} \) |
| 19 | \( 1 + 7.77e19T + 4.37e39T^{2} \) |
| 23 | \( 1 + 1.47e21T + 1.63e42T^{2} \) |
| 29 | \( 1 + 4.71e22T + 2.15e45T^{2} \) |
| 31 | \( 1 - 1.23e23T + 1.70e46T^{2} \) |
| 37 | \( 1 + 6.07e23T + 4.11e48T^{2} \) |
| 41 | \( 1 + 1.14e25T + 9.91e49T^{2} \) |
| 43 | \( 1 + 2.03e24T + 4.34e50T^{2} \) |
| 47 | \( 1 - 9.37e25T + 6.83e51T^{2} \) |
| 53 | \( 1 - 2.80e26T + 2.83e53T^{2} \) |
| 59 | \( 1 + 1.73e27T + 7.87e54T^{2} \) |
| 61 | \( 1 + 3.03e26T + 2.21e55T^{2} \) |
| 67 | \( 1 - 1.97e28T + 4.05e56T^{2} \) |
| 71 | \( 1 - 6.31e28T + 2.44e57T^{2} \) |
| 73 | \( 1 - 6.77e28T + 5.79e57T^{2} \) |
| 79 | \( 1 - 1.91e29T + 6.70e58T^{2} \) |
| 83 | \( 1 + 4.70e27T + 3.10e59T^{2} \) |
| 89 | \( 1 - 3.89e29T + 2.69e60T^{2} \) |
| 97 | \( 1 - 2.01e30T + 3.88e61T^{2} \) |
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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
−20.01105660898587724230094696828, −18.77814365731435503504062121491, −16.70058092187662715182165716535, −15.58346985013191643414423181190, −12.11788598820107532506736370378, −10.94959143486583464217412494644, −8.389213250405636643321970553291, −6.48029438036995491002628402012, −3.72883487562511292311851567497, −0.48347077370412797938556807682,
0.48347077370412797938556807682, 3.72883487562511292311851567497, 6.48029438036995491002628402012, 8.389213250405636643321970553291, 10.94959143486583464217412494644, 12.11788598820107532506736370378, 15.58346985013191643414423181190, 16.70058092187662715182165716535, 18.77814365731435503504062121491, 20.01105660898587724230094696828
