| L(s) = 1 | + 9·3-s + 65.5·5-s + 81·9-s − 245.·11-s + 434.·13-s + 590.·15-s − 1.10e3·17-s − 2.87e3·19-s − 4.22e3·23-s + 1.17e3·25-s + 729·27-s + 4.96e3·29-s − 8.78e3·31-s − 2.21e3·33-s − 2.44e3·37-s + 3.91e3·39-s − 3.66e3·41-s − 7.19e3·43-s + 5.31e3·45-s + 3.27e3·47-s − 9.92e3·51-s − 3.02e3·53-s − 1.61e4·55-s − 2.58e4·57-s − 5.14e4·59-s − 1.33e4·61-s + 2.85e4·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.17·5-s + 0.333·9-s − 0.611·11-s + 0.713·13-s + 0.677·15-s − 0.925·17-s − 1.82·19-s − 1.66·23-s + 0.375·25-s + 0.192·27-s + 1.09·29-s − 1.64·31-s − 0.353·33-s − 0.293·37-s + 0.412·39-s − 0.340·41-s − 0.593·43-s + 0.391·45-s + 0.216·47-s − 0.534·51-s − 0.147·53-s − 0.717·55-s − 1.05·57-s − 1.92·59-s − 0.458·61-s + 0.837·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - 9T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 65.5T + 3.12e3T^{2} \) |
| 11 | \( 1 + 245.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 434.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.10e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.87e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.22e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.96e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 8.78e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.44e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 3.66e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 7.19e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 3.27e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.02e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 5.14e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.33e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.08e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.18e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.46e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.75e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.00e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.07e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.40e5T + 8.58e9T^{2} \) |
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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
−9.402108979184850373613484773864, −8.671521369871877751444029150487, −7.898167524291616087080901357767, −6.55529239654547634993881089896, −6.00914371027962963478502220060, −4.78133924744764873514058081633, −3.70060304537933683646842776416, −2.31172643502052337788849761468, −1.78400884788701691630491324446, 0,
1.78400884788701691630491324446, 2.31172643502052337788849761468, 3.70060304537933683646842776416, 4.78133924744764873514058081633, 6.00914371027962963478502220060, 6.55529239654547634993881089896, 7.898167524291616087080901357767, 8.671521369871877751444029150487, 9.402108979184850373613484773864
