| L(s) = 1 | − 5.31·2-s + 9·3-s − 3.71·4-s + 103.·5-s − 47.8·6-s + 189.·8-s + 81·9-s − 550.·10-s + 653.·11-s − 33.4·12-s − 138.·13-s + 931.·15-s − 891.·16-s + 1.17e3·17-s − 430.·18-s − 1.71e3·19-s − 384.·20-s − 3.47e3·22-s − 4.02e3·23-s + 1.70e3·24-s + 7.58e3·25-s + 734.·26-s + 729·27-s + 2.64e3·29-s − 4.95e3·30-s − 2.87e3·31-s − 1.33e3·32-s + ⋯ |
| L(s) = 1 | − 0.940·2-s + 0.577·3-s − 0.116·4-s + 1.85·5-s − 0.542·6-s + 1.04·8-s + 0.333·9-s − 1.74·10-s + 1.62·11-s − 0.0670·12-s − 0.226·13-s + 1.06·15-s − 0.870·16-s + 0.985·17-s − 0.313·18-s − 1.08·19-s − 0.215·20-s − 1.53·22-s − 1.58·23-s + 0.605·24-s + 2.42·25-s + 0.212·26-s + 0.192·27-s + 0.585·29-s − 1.00·30-s − 0.537·31-s − 0.231·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.041737097\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.041737097\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 5.31T + 32T^{2} \) |
| 5 | \( 1 - 103.T + 3.12e3T^{2} \) |
| 11 | \( 1 - 653.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 138.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.17e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.71e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.02e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.64e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.87e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.85e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 216.T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.92e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.48e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.11e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.46e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 8.75e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.20e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.55e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.34e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.31e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.33e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.03e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.62e4T + 8.58e9T^{2} \) |
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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
−12.24021461025367975578381086043, −10.52854891473395275107743391318, −9.801520685781124159301607285382, −9.212669589709511611108954473714, −8.349820473367381836486921658888, −6.87817671375199741233615366752, −5.76759868748204443316294305813, −4.14312150386472702852850793792, −2.15498759034873486040431736349, −1.19969189940168022109989643484,
1.19969189940168022109989643484, 2.15498759034873486040431736349, 4.14312150386472702852850793792, 5.76759868748204443316294305813, 6.87817671375199741233615366752, 8.349820473367381836486921658888, 9.212669589709511611108954473714, 9.801520685781124159301607285382, 10.52854891473395275107743391318, 12.24021461025367975578381086043
