| L(s) = 1 | + 0.226·2-s − 0.219·3-s − 1.94·4-s − 2.49·5-s − 0.0495·6-s + 7-s − 0.893·8-s − 2.95·9-s − 0.564·10-s + 0.427·12-s + 5.13·13-s + 0.226·14-s + 0.546·15-s + 3.69·16-s − 1.43·17-s − 0.667·18-s − 6.06·19-s + 4.86·20-s − 0.219·21-s + 7.08·23-s + 0.195·24-s + 1.22·25-s + 1.16·26-s + 1.30·27-s − 1.94·28-s + 6.51·29-s + 0.123·30-s + ⋯ |
| L(s) = 1 | + 0.159·2-s − 0.126·3-s − 0.974·4-s − 1.11·5-s − 0.0202·6-s + 0.377·7-s − 0.315·8-s − 0.983·9-s − 0.178·10-s + 0.123·12-s + 1.42·13-s + 0.0604·14-s + 0.141·15-s + 0.923·16-s − 0.348·17-s − 0.157·18-s − 1.39·19-s + 1.08·20-s − 0.0478·21-s + 1.47·23-s + 0.0399·24-s + 0.245·25-s + 0.227·26-s + 0.251·27-s − 0.368·28-s + 1.21·29-s + 0.0225·30-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
| \(L(1)\) |
\(\approx\) |
\(0.9126058196\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9126058196\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.226T + 2T^{2} \) |
| 3 | \( 1 + 0.219T + 3T^{2} \) |
| 5 | \( 1 + 2.49T + 5T^{2} \) |
| 13 | \( 1 - 5.13T + 13T^{2} \) |
| 17 | \( 1 + 1.43T + 17T^{2} \) |
| 19 | \( 1 + 6.06T + 19T^{2} \) |
| 23 | \( 1 - 7.08T + 23T^{2} \) |
| 29 | \( 1 - 6.51T + 29T^{2} \) |
| 31 | \( 1 - 7.68T + 31T^{2} \) |
| 37 | \( 1 + 3.98T + 37T^{2} \) |
| 41 | \( 1 - 6.74T + 41T^{2} \) |
| 43 | \( 1 + 0.802T + 43T^{2} \) |
| 47 | \( 1 - 6.75T + 47T^{2} \) |
| 53 | \( 1 - 6.58T + 53T^{2} \) |
| 59 | \( 1 - 2.87T + 59T^{2} \) |
| 61 | \( 1 + 0.855T + 61T^{2} \) |
| 67 | \( 1 + 1.64T + 67T^{2} \) |
| 71 | \( 1 - 4.52T + 71T^{2} \) |
| 73 | \( 1 + 14.8T + 73T^{2} \) |
| 79 | \( 1 - 2.45T + 79T^{2} \) |
| 83 | \( 1 - 2.24T + 83T^{2} \) |
| 89 | \( 1 - 1.73T + 89T^{2} \) |
| 97 | \( 1 + 12.0T + 97T^{2} \) |
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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
−10.40824592639988316509360686159, −8.878967655441479145673798793928, −8.628224968133542046979207737845, −7.957479777243758773904675020962, −6.64679000568740930643517737013, −5.72079077955239506445624912989, −4.63077126619043036104200470965, −3.98727601889932024494652120202, −2.92198446594862392218162626649, −0.76670312481976986468045789396,
0.76670312481976986468045789396, 2.92198446594862392218162626649, 3.98727601889932024494652120202, 4.63077126619043036104200470965, 5.72079077955239506445624912989, 6.64679000568740930643517737013, 7.957479777243758773904675020962, 8.628224968133542046979207737845, 8.878967655441479145673798793928, 10.40824592639988316509360686159
