| L(s) = 1 | − 1.90·2-s − 9.91·3-s − 4.36·4-s − 19.7·5-s + 18.9·6-s + 7·7-s + 23.5·8-s + 71.3·9-s + 37.7·10-s + 43.2·12-s + 21.7·13-s − 13.3·14-s + 196.·15-s − 10.0·16-s + 9.40·17-s − 136.·18-s − 123.·19-s + 86.3·20-s − 69.4·21-s − 66.2·23-s − 233.·24-s + 266.·25-s − 41.4·26-s − 440.·27-s − 30.5·28-s − 91.3·29-s − 374.·30-s + ⋯ |
| L(s) = 1 | − 0.674·2-s − 1.90·3-s − 0.545·4-s − 1.76·5-s + 1.28·6-s + 0.377·7-s + 1.04·8-s + 2.64·9-s + 1.19·10-s + 1.04·12-s + 0.463·13-s − 0.254·14-s + 3.37·15-s − 0.156·16-s + 0.134·17-s − 1.78·18-s − 1.48·19-s + 0.965·20-s − 0.721·21-s − 0.600·23-s − 1.98·24-s + 2.13·25-s − 0.312·26-s − 3.13·27-s − 0.206·28-s − 0.585·29-s − 2.27·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1095206589\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1095206589\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 7T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.90T + 8T^{2} \) |
| 3 | \( 1 + 9.91T + 27T^{2} \) |
| 5 | \( 1 + 19.7T + 125T^{2} \) |
| 13 | \( 1 - 21.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 9.40T + 4.91e3T^{2} \) |
| 19 | \( 1 + 123.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 66.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 91.3T + 2.43e4T^{2} \) |
| 31 | \( 1 - 34.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 163.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 480.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 236.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 147.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 497.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 327.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 803.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 2.41T + 3.00e5T^{2} \) |
| 71 | \( 1 - 6.07T + 3.57e5T^{2} \) |
| 73 | \( 1 + 255.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 84.6T + 4.93e5T^{2} \) |
| 83 | \( 1 - 22.1T + 5.71e5T^{2} \) |
| 89 | \( 1 + 983.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 133.T + 9.12e5T^{2} \) |
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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
−10.08936567826956558876322541149, −8.868194969639942097641975877009, −7.990753954868660366091103750190, −7.36104611161194480477015669886, −6.40974141605113467450076758220, −5.29482752814938441287103733633, −4.33859137813257697358504690864, −4.00675610041771165995291184378, −1.35341419267324075380845455100, −0.25210129239715520926142403824,
0.25210129239715520926142403824, 1.35341419267324075380845455100, 4.00675610041771165995291184378, 4.33859137813257697358504690864, 5.29482752814938441287103733633, 6.40974141605113467450076758220, 7.36104611161194480477015669886, 7.990753954868660366091103750190, 8.868194969639942097641975877009, 10.08936567826956558876322541149
