| L(s) = 1 | − 1.53·2-s + 10.1·3-s − 5.65·4-s + 8.69·5-s − 15.5·6-s + 7·7-s + 20.9·8-s + 75.9·9-s − 13.3·10-s − 57.3·12-s + 76.3·13-s − 10.7·14-s + 88.2·15-s + 13.1·16-s − 39.7·17-s − 116.·18-s + 27.9·19-s − 49.1·20-s + 71.0·21-s + 87.2·23-s + 212.·24-s − 49.3·25-s − 117.·26-s + 496.·27-s − 39.5·28-s + 38.3·29-s − 135.·30-s + ⋯ |
| L(s) = 1 | − 0.541·2-s + 1.95·3-s − 0.706·4-s + 0.778·5-s − 1.05·6-s + 0.377·7-s + 0.924·8-s + 2.81·9-s − 0.421·10-s − 1.37·12-s + 1.62·13-s − 0.204·14-s + 1.51·15-s + 0.205·16-s − 0.566·17-s − 1.52·18-s + 0.337·19-s − 0.549·20-s + 0.738·21-s + 0.790·23-s + 1.80·24-s − 0.394·25-s − 0.882·26-s + 3.53·27-s − 0.267·28-s + 0.245·29-s − 0.823·30-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
| \(L(2)\) |
\(\approx\) |
\(3.823603788\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.823603788\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 1.53T + 8T^{2} \) |
| 3 | \( 1 - 10.1T + 27T^{2} \) |
| 5 | \( 1 - 8.69T + 125T^{2} \) |
| 13 | \( 1 - 76.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 39.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 27.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 87.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 38.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + 186.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 218.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 80.1T + 6.89e4T^{2} \) |
| 43 | \( 1 - 35.1T + 7.95e4T^{2} \) |
| 47 | \( 1 + 282.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 145.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 91.0T + 2.05e5T^{2} \) |
| 61 | \( 1 + 808.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 794.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 946.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 801.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 890.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 559.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.52e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 664.T + 9.12e5T^{2} \) |
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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
−9.436382949938739328894601286004, −8.937017565770596399939699497486, −8.402967354768908618125773366292, −7.66892548774352360557371849003, −6.65834063268351398515170450865, −5.20006468332940115647370405781, −4.08032869886432579776275979658, −3.32745693309031466662825034179, −1.96975157668850305351477473474, −1.24961415042202564201619500976,
1.24961415042202564201619500976, 1.96975157668850305351477473474, 3.32745693309031466662825034179, 4.08032869886432579776275979658, 5.20006468332940115647370405781, 6.65834063268351398515170450865, 7.66892548774352360557371849003, 8.402967354768908618125773366292, 8.937017565770596399939699497486, 9.436382949938739328894601286004
