| L(s) = 1 | + 7.66·2-s + 9·3-s + 26.8·4-s − 109.·5-s + 69.0·6-s + 156.·7-s − 39.7·8-s + 81·9-s − 842.·10-s − 426.·11-s + 241.·12-s − 123.·13-s + 1.19e3·14-s − 988.·15-s − 1.16e3·16-s − 852.·17-s + 621.·18-s + 232.·19-s − 2.94e3·20-s + 1.40e3·21-s − 3.27e3·22-s − 3.64e3·23-s − 358.·24-s + 8.93e3·25-s − 950.·26-s + 729·27-s + 4.19e3·28-s + ⋯ |
| L(s) = 1 | + 1.35·2-s + 0.577·3-s + 0.837·4-s − 1.96·5-s + 0.782·6-s + 1.20·7-s − 0.219·8-s + 0.333·9-s − 2.66·10-s − 1.06·11-s + 0.483·12-s − 0.203·13-s + 1.63·14-s − 1.13·15-s − 1.13·16-s − 0.715·17-s + 0.451·18-s + 0.147·19-s − 1.64·20-s + 0.696·21-s − 1.44·22-s − 1.43·23-s − 0.126·24-s + 2.85·25-s − 0.275·26-s + 0.192·27-s + 1.01·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 9T \) |
| 59 | \( 1 - 3.48e3T \) |
| good | 2 | \( 1 - 7.66T + 32T^{2} \) |
| 5 | \( 1 + 109.T + 3.12e3T^{2} \) |
| 7 | \( 1 - 156.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 426.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 123.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 852.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 232.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.64e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 8.17e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.35e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.31e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.30e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.33e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 8.86e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.30e4T + 4.18e8T^{2} \) |
| 61 | \( 1 - 1.36e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.42e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.45e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.01e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.97e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.98e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.07e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 2.19e4T + 8.58e9T^{2} \) |
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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
−11.44608214516480729728436362633, −10.95844993477036685560307926854, −8.960384777370689026828636767186, −7.86424574590455072330140316334, −7.39068443132225472405953006552, −5.46992118620934781966119167162, −4.36042903086877353026528197734, −3.81018393958993599091467508011, −2.40083319562865224653682549025, 0,
2.40083319562865224653682549025, 3.81018393958993599091467508011, 4.36042903086877353026528197734, 5.46992118620934781966119167162, 7.39068443132225472405953006552, 7.86424574590455072330140316334, 8.960384777370689026828636767186, 10.95844993477036685560307926854, 11.44608214516480729728436362633
