| L(s) = 1 | + 7.00·2-s − 78.8·4-s − 449.·5-s − 271.·7-s − 1.44e3·8-s − 3.14e3·10-s + 3.16e3·11-s − 9.44e3·13-s − 1.90e3·14-s − 63.3·16-s + 6.79e3·17-s + 3.40e4·19-s + 3.54e4·20-s + 2.21e4·22-s + 3.12e4·23-s + 1.23e5·25-s − 6.62e4·26-s + 2.14e4·28-s − 8.27e4·29-s + 2.83e5·31-s + 1.85e5·32-s + 4.76e4·34-s + 1.22e5·35-s + 2.68e4·37-s + 2.38e5·38-s + 6.51e5·40-s + 1.95e4·41-s + ⋯ |
| L(s) = 1 | + 0.619·2-s − 0.616·4-s − 1.60·5-s − 0.299·7-s − 1.00·8-s − 0.995·10-s + 0.716·11-s − 1.19·13-s − 0.185·14-s − 0.00386·16-s + 0.335·17-s + 1.14·19-s + 0.990·20-s + 0.444·22-s + 0.535·23-s + 1.58·25-s − 0.738·26-s + 0.184·28-s − 0.629·29-s + 1.71·31-s + 0.998·32-s + 0.207·34-s + 0.481·35-s + 0.0870·37-s + 0.706·38-s + 1.60·40-s + 0.0442·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 + 2.05e5T \) |
| good | 2 | \( 1 - 7.00T + 128T^{2} \) |
| 5 | \( 1 + 449.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 271.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 3.16e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 9.44e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 6.79e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.40e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.12e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 8.27e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.83e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.68e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 1.95e4T + 1.94e11T^{2} \) |
| 43 | \( 1 + 4.17e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 2.76e4T + 5.06e11T^{2} \) |
| 53 | \( 1 + 2.90e4T + 1.17e12T^{2} \) |
| 61 | \( 1 - 1.53e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.52e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.35e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.90e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.26e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.96e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 5.11e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.45e5T + 8.07e13T^{2} \) |
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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
−9.274309271539556676041212877930, −8.267131455545471495181568031498, −7.50050650947884286878573465574, −6.56122538244264284551418861312, −5.21302863560028495294584469692, −4.50230478090456652158841057906, −3.62568141934094170323956424917, −2.92601982474903406575168629572, −0.913253144127034297947856632617, 0,
0.913253144127034297947856632617, 2.92601982474903406575168629572, 3.62568141934094170323956424917, 4.50230478090456652158841057906, 5.21302863560028495294584469692, 6.56122538244264284551418861312, 7.50050650947884286878573465574, 8.267131455545471495181568031498, 9.274309271539556676041212877930
