| L(s) = 1 | − 1.40·2-s − 2.16·3-s − 0.0139·4-s − 1.83·5-s + 3.04·6-s + 7-s + 2.83·8-s + 1.68·9-s + 2.58·10-s + 0.0302·12-s − 4.64·13-s − 1.40·14-s + 3.96·15-s − 3.97·16-s − 5.47·17-s − 2.36·18-s − 5.80·19-s + 0.0255·20-s − 2.16·21-s − 0.719·23-s − 6.14·24-s − 1.64·25-s + 6.54·26-s + 2.85·27-s − 0.0139·28-s − 1.17·29-s − 5.58·30-s + ⋯ |
| L(s) = 1 | − 0.996·2-s − 1.24·3-s − 0.00698·4-s − 0.819·5-s + 1.24·6-s + 0.377·7-s + 1.00·8-s + 0.560·9-s + 0.816·10-s + 0.00872·12-s − 1.28·13-s − 0.376·14-s + 1.02·15-s − 0.992·16-s − 1.32·17-s − 0.558·18-s − 1.33·19-s + 0.00571·20-s − 0.472·21-s − 0.150·23-s − 1.25·24-s − 0.329·25-s + 1.28·26-s + 0.549·27-s − 0.00263·28-s − 0.217·29-s − 1.01·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1871410340\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1871410340\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.40T + 2T^{2} \) |
| 3 | \( 1 + 2.16T + 3T^{2} \) |
| 5 | \( 1 + 1.83T + 5T^{2} \) |
| 13 | \( 1 + 4.64T + 13T^{2} \) |
| 17 | \( 1 + 5.47T + 17T^{2} \) |
| 19 | \( 1 + 5.80T + 19T^{2} \) |
| 23 | \( 1 + 0.719T + 23T^{2} \) |
| 29 | \( 1 + 1.17T + 29T^{2} \) |
| 31 | \( 1 + 1.30T + 31T^{2} \) |
| 37 | \( 1 - 2.09T + 37T^{2} \) |
| 41 | \( 1 + 0.916T + 41T^{2} \) |
| 43 | \( 1 - 8.02T + 43T^{2} \) |
| 47 | \( 1 - 5.97T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 + 7.68T + 59T^{2} \) |
| 61 | \( 1 + 6.27T + 61T^{2} \) |
| 67 | \( 1 + 15.4T + 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 - 6.01T + 73T^{2} \) |
| 79 | \( 1 + 15.6T + 79T^{2} \) |
| 83 | \( 1 + 4.37T + 83T^{2} \) |
| 89 | \( 1 - 15.3T + 89T^{2} \) |
| 97 | \( 1 - 2.41T + 97T^{2} \) |
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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
−10.42129187109114356327345317966, −9.298589278585516075187414555801, −8.561878701653644827222606762895, −7.64306275942812115865299248208, −6.98841735172561589048140142094, −5.88572355226550738192600933050, −4.69157929626564479419555310499, −4.26685292618576175685198773340, −2.16901827663704131159076979811, −0.41389004845033337378477967177,
0.41389004845033337378477967177, 2.16901827663704131159076979811, 4.26685292618576175685198773340, 4.69157929626564479419555310499, 5.88572355226550738192600933050, 6.98841735172561589048140142094, 7.64306275942812115865299248208, 8.561878701653644827222606762895, 9.298589278585516075187414555801, 10.42129187109114356327345317966
