| L(s) = 1 | + 5.09·2-s + 17.9·4-s − 18.5·5-s + 50.5·8-s − 94.7·10-s − 1.05·11-s − 58.7·13-s + 113.·16-s + 43.7·17-s + 131.·19-s − 333.·20-s − 5.37·22-s + 161.·23-s + 220.·25-s − 299.·26-s + 64.0·29-s + 55.9·31-s + 175.·32-s + 222.·34-s + 296.·37-s + 671.·38-s − 940.·40-s + 80.6·41-s − 134.·43-s − 18.9·44-s + 821.·46-s + 233.·47-s + ⋯ |
| L(s) = 1 | + 1.80·2-s + 2.24·4-s − 1.66·5-s + 2.23·8-s − 2.99·10-s − 0.0289·11-s − 1.25·13-s + 1.78·16-s + 0.624·17-s + 1.59·19-s − 3.72·20-s − 0.0521·22-s + 1.46·23-s + 1.76·25-s − 2.25·26-s + 0.409·29-s + 0.324·31-s + 0.971·32-s + 1.12·34-s + 1.31·37-s + 2.86·38-s − 3.71·40-s + 0.307·41-s − 0.476·43-s − 0.0648·44-s + 2.63·46-s + 0.724·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.151679682\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.151679682\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 5.09T + 8T^{2} \) |
| 5 | \( 1 + 18.5T + 125T^{2} \) |
| 11 | \( 1 + 1.05T + 1.33e3T^{2} \) |
| 13 | \( 1 + 58.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 43.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 131.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 161.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 64.0T + 2.43e4T^{2} \) |
| 31 | \( 1 - 55.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 296.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 80.6T + 6.89e4T^{2} \) |
| 43 | \( 1 + 134.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 233.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 387.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 722.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 388.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 730.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 685.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 275.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 854.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 922.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 301.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 913.T + 9.12e5T^{2} \) |
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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
−9.282517451975435014638674012148, −7.950882383811106001454609044179, −7.38014733470074022132095539442, −6.85270039733057877670900319529, −5.55184639852496652550641973113, −4.87613024791808590327393997607, −4.19530409908093360024800364369, −3.26444592284693035249632935752, −2.71002204849745509345082157547, −0.876891727874681402328249459012,
0.876891727874681402328249459012, 2.71002204849745509345082157547, 3.26444592284693035249632935752, 4.19530409908093360024800364369, 4.87613024791808590327393997607, 5.55184639852496652550641973113, 6.85270039733057877670900319529, 7.38014733470074022132095539442, 7.950882383811106001454609044179, 9.282517451975435014638674012148
