L(s) = 1 | − 4·2-s − 21·3-s + 16·4-s + 81·5-s + 84·6-s + 98·7-s − 64·8-s + 198·9-s − 324·10-s + 121·11-s − 336·12-s + 824·13-s − 392·14-s − 1.70e3·15-s + 256·16-s + 978·17-s − 792·18-s − 2.14e3·19-s + 1.29e3·20-s − 2.05e3·21-s − 484·22-s + 3.69e3·23-s + 1.34e3·24-s + 3.43e3·25-s − 3.29e3·26-s + 945·27-s + 1.56e3·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.34·3-s + 1/2·4-s + 1.44·5-s + 0.952·6-s + 0.755·7-s − 0.353·8-s + 0.814·9-s − 1.02·10-s + 0.301·11-s − 0.673·12-s + 1.35·13-s − 0.534·14-s − 1.95·15-s + 1/4·16-s + 0.820·17-s − 0.576·18-s − 1.35·19-s + 0.724·20-s − 1.01·21-s − 0.213·22-s + 1.45·23-s + 0.476·24-s + 1.09·25-s − 0.956·26-s + 0.249·27-s + 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.9406718784\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9406718784\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + p^{2} T \) |
| 11 | \( 1 - p^{2} T \) |
good | 3 | \( 1 + 7 p T + p^{5} T^{2} \) |
| 5 | \( 1 - 81 T + p^{5} T^{2} \) |
| 7 | \( 1 - 2 p^{2} T + p^{5} T^{2} \) |
| 13 | \( 1 - 824 T + p^{5} T^{2} \) |
| 17 | \( 1 - 978 T + p^{5} T^{2} \) |
| 19 | \( 1 + 2140 T + p^{5} T^{2} \) |
| 23 | \( 1 - 3699 T + p^{5} T^{2} \) |
| 29 | \( 1 - 120 p T + p^{5} T^{2} \) |
| 31 | \( 1 + 7813 T + p^{5} T^{2} \) |
| 37 | \( 1 + 13597 T + p^{5} T^{2} \) |
| 41 | \( 1 - 6492 T + p^{5} T^{2} \) |
| 43 | \( 1 - 14234 T + p^{5} T^{2} \) |
| 47 | \( 1 + 20352 T + p^{5} T^{2} \) |
| 53 | \( 1 + 366 T + p^{5} T^{2} \) |
| 59 | \( 1 - 9825 T + p^{5} T^{2} \) |
| 61 | \( 1 - 26132 T + p^{5} T^{2} \) |
| 67 | \( 1 - 17093 T + p^{5} T^{2} \) |
| 71 | \( 1 + 23583 T + p^{5} T^{2} \) |
| 73 | \( 1 + 35176 T + p^{5} T^{2} \) |
| 79 | \( 1 + 42490 T + p^{5} T^{2} \) |
| 83 | \( 1 - 22674 T + p^{5} T^{2} \) |
| 89 | \( 1 + 17145 T + p^{5} T^{2} \) |
| 97 | \( 1 + 30727 T + p^{5} T^{2} \) |
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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
−17.32068231276180834664516858820, −16.29548411569198450994408504892, −14.45847546651427787147018449759, −12.83217645361744585731682355329, −11.22757978927652857014871325453, −10.44211558850650432678746479523, −8.821410682964445825677633750722, −6.54068594782954764725318890123, −5.43962434674073784085121528752, −1.38413119829197160596611746929,
1.38413119829197160596611746929, 5.43962434674073784085121528752, 6.54068594782954764725318890123, 8.821410682964445825677633750722, 10.44211558850650432678746479523, 11.22757978927652857014871325453, 12.83217645361744585731682355329, 14.45847546651427787147018449759, 16.29548411569198450994408504892, 17.32068231276180834664516858820