| L(s) = 1 | − 3·3-s − 7.63·5-s + 5.63·7-s + 9·9-s + 34.5·11-s + 13·13-s + 22.8·15-s + 2·17-s − 88.1·19-s − 16.8·21-s − 64·23-s − 66.7·25-s − 27·27-s + 23.7·29-s − 284.·31-s − 103.·33-s − 42.9·35-s + 115.·37-s − 39·39-s + 1.41·41-s − 337.·43-s − 68.6·45-s − 198.·47-s − 311.·49-s − 6·51-s + 59.0·53-s − 263.·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.682·5-s + 0.303·7-s + 0.333·9-s + 0.946·11-s + 0.277·13-s + 0.394·15-s + 0.0285·17-s − 1.06·19-s − 0.175·21-s − 0.580·23-s − 0.534·25-s − 0.192·27-s + 0.152·29-s − 1.64·31-s − 0.546·33-s − 0.207·35-s + 0.512·37-s − 0.160·39-s + 0.00537·41-s − 1.19·43-s − 0.227·45-s − 0.615·47-s − 0.907·49-s − 0.0164·51-s + 0.153·53-s − 0.645·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 3T \) |
| 13 | \( 1 - 13T \) |
| good | 5 | \( 1 + 7.63T + 125T^{2} \) |
| 7 | \( 1 - 5.63T + 343T^{2} \) |
| 11 | \( 1 - 34.5T + 1.33e3T^{2} \) |
| 17 | \( 1 - 2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 88.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 64T + 1.21e4T^{2} \) |
| 29 | \( 1 - 23.7T + 2.43e4T^{2} \) |
| 31 | \( 1 + 284.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 115.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 1.41T + 6.89e4T^{2} \) |
| 43 | \( 1 + 337.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 198.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 59.0T + 1.48e5T^{2} \) |
| 59 | \( 1 + 188.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 336.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 531.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 510.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 164.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 29.3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 117.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 508.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.02e3T + 9.12e5T^{2} \) |
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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
−11.02978616120094852203988797375, −9.940840277987285178439966104772, −8.825879345907695581214683451340, −7.87759948175251218017594841934, −6.80782589689300687629677522991, −5.86677578586460800073622475365, −4.54013886905400691343241210497, −3.65589810597696841341480919043, −1.68543225700646612433393461335, 0,
1.68543225700646612433393461335, 3.65589810597696841341480919043, 4.54013886905400691343241210497, 5.86677578586460800073622475365, 6.80782589689300687629677522991, 7.87759948175251218017594841934, 8.825879345907695581214683451340, 9.940840277987285178439966104772, 11.02978616120094852203988797375
