| L(s) = 1 | − 5.21·2-s − 100.·4-s − 125·5-s + 343·7-s + 1.19e3·8-s + 651.·10-s − 7.40e3·11-s + 3.78e3·13-s − 1.78e3·14-s + 6.68e3·16-s − 1.92e4·17-s − 1.66e4·19-s + 1.26e4·20-s + 3.85e4·22-s + 3.19e4·23-s + 1.56e4·25-s − 1.97e4·26-s − 3.45e4·28-s − 4.53e4·29-s + 9.24e4·31-s − 1.87e5·32-s + 1.00e5·34-s − 4.28e4·35-s − 2.66e5·37-s + 8.66e4·38-s − 1.49e5·40-s − 6.27e5·41-s + ⋯ |
| L(s) = 1 | − 0.460·2-s − 0.787·4-s − 0.447·5-s + 0.377·7-s + 0.823·8-s + 0.206·10-s − 1.67·11-s + 0.478·13-s − 0.174·14-s + 0.408·16-s − 0.949·17-s − 0.556·19-s + 0.352·20-s + 0.772·22-s + 0.548·23-s + 0.199·25-s − 0.220·26-s − 0.297·28-s − 0.345·29-s + 0.557·31-s − 1.01·32-s + 0.437·34-s − 0.169·35-s − 0.864·37-s + 0.256·38-s − 0.368·40-s − 1.42·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.5142849777\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5142849777\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 125T \) |
| 7 | \( 1 - 343T \) |
| good | 2 | \( 1 + 5.21T + 128T^{2} \) |
| 11 | \( 1 + 7.40e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 3.78e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.92e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.66e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.19e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 4.53e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 9.24e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.66e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 6.27e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 5.57e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 4.12e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.34e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.79e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.92e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.92e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.95e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.95e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 7.19e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 7.88e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 3.37e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 9.18e6T + 8.07e13T^{2} \) |
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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
−10.54712230227338936503792395099, −9.431286586748790640882365613695, −8.356798125329062848803055674627, −8.017783119285445173584636375888, −6.76692045192051786714648991866, −5.24156645001439131039576441254, −4.58934992680219638949458051319, −3.29773272498839059016604499672, −1.82915024952000822792936361249, −0.36877315015844074658716197114,
0.36877315015844074658716197114, 1.82915024952000822792936361249, 3.29773272498839059016604499672, 4.58934992680219638949458051319, 5.24156645001439131039576441254, 6.76692045192051786714648991866, 8.017783119285445173584636375888, 8.356798125329062848803055674627, 9.431286586748790640882365613695, 10.54712230227338936503792395099
