| L(s) = 1 | + 8.20·2-s − 9·3-s + 35.3·4-s − 29.2·5-s − 73.8·6-s + 27.7·8-s + 81·9-s − 239.·10-s + 377.·11-s − 318.·12-s − 509.·13-s + 263.·15-s − 904.·16-s − 1.60e3·17-s + 664.·18-s − 2.15e3·19-s − 1.03e3·20-s + 3.09e3·22-s − 4.43e3·23-s − 249.·24-s − 2.27e3·25-s − 4.18e3·26-s − 729·27-s + 4.77e3·29-s + 2.15e3·30-s + 7.15e3·31-s − 8.31e3·32-s + ⋯ |
| L(s) = 1 | + 1.45·2-s − 0.577·3-s + 1.10·4-s − 0.522·5-s − 0.837·6-s + 0.153·8-s + 0.333·9-s − 0.758·10-s + 0.940·11-s − 0.638·12-s − 0.836·13-s + 0.301·15-s − 0.883·16-s − 1.34·17-s + 0.483·18-s − 1.37·19-s − 0.578·20-s + 1.36·22-s − 1.74·23-s − 0.0885·24-s − 0.726·25-s − 1.21·26-s − 0.192·27-s + 1.05·29-s + 0.438·30-s + 1.33·31-s − 1.43·32-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
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 9T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 8.20T + 32T^{2} \) |
| 5 | \( 1 + 29.2T + 3.12e3T^{2} \) |
| 11 | \( 1 - 377.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 509.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.60e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.15e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.43e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.77e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.15e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.57e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 9.91e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 4.88e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 9.26e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 9.24e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.41e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.80e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.65e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.42e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.09e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.14e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.15e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.82e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.51e5T + 8.58e9T^{2} \) |
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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
−11.91860831090824998288161558952, −11.16162988087801019605954664455, −9.789549042264125578219781577615, −8.314190925248922306875427547966, −6.73840866217908018437403517007, −6.09485648919186272871895226075, −4.56359145091713755426172115937, −4.07658150880127095961015263554, −2.31059951965208552463244263345, 0,
2.31059951965208552463244263345, 4.07658150880127095961015263554, 4.56359145091713755426172115937, 6.09485648919186272871895226075, 6.73840866217908018437403517007, 8.314190925248922306875427547966, 9.789549042264125578219781577615, 11.16162988087801019605954664455, 11.91860831090824998288161558952
