L(s) = 1 | − 1.41·2-s + 2.00·4-s − 4.46i·5-s − 2.82·8-s + 6.30i·10-s + 5.99i·13-s + 4.00·16-s + 4.90i·17-s − 8.92i·20-s − 14.8·25-s − 8.47i·26-s − 4.24·29-s − 5.65·32-s − 6.94i·34-s − 9.89·37-s + ⋯ |
L(s) = 1 | − 1.00·2-s + 1.00·4-s − 1.99i·5-s − 1.00·8-s + 1.99i·10-s + 1.66i·13-s + 1.00·16-s + 1.19i·17-s − 1.99i·20-s − 2.97·25-s − 1.66i·26-s − 0.787·29-s − 1.00·32-s − 1.19i·34-s − 1.62·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.156 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.156 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3244458205\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3244458205\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 4.46iT - 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 5.99iT - 13T^{2} \) |
| 17 | \( 1 - 4.90iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 4.24T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 9.89T + 37T^{2} \) |
| 41 | \( 1 - 3.56iT - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 14T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 7.25iT - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 11.6iT - 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 3.11iT - 89T^{2} \) |
| 97 | \( 1 + 13.5iT - 97T^{2} \) |
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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
−9.356554060367622368000253942561, −8.711258880341893226247718626496, −8.305912971010849828366233237750, −7.37124960891758479558074730343, −6.37674509318586288949630974728, −5.56601353717950790132646035570, −4.58421610699766338929106211156, −3.73460939411724930315780139620, −1.92154934375493379713119539535, −1.37804314961950780359929871711,
0.16461435941883506991208667661, 2.03410388278795894468886233669, 3.02624746462146134096714794570, 3.42063437193610336860236007917, 5.32604597470365176931109063674, 6.15958295492554414862253100012, 6.92909844142046773348769744407, 7.52124561352620427516376049835, 8.069673213702405694465189707282, 9.260479145174871549266686026321