L(s) = 1 | + 28.9i·3-s + (13.1 + 54.3i)5-s + 146. i·7-s − 594.·9-s − 191.·11-s + 83.9i·13-s + (−1.57e3 + 380. i)15-s − 2.00e3i·17-s + 677.·19-s − 4.24e3·21-s − 1.29e3i·23-s + (−2.77e3 + 1.42e3i)25-s − 1.01e4i·27-s − 3.26e3·29-s + 6.15e3·31-s + ⋯ |
L(s) = 1 | + 1.85i·3-s + (0.235 + 0.971i)5-s + 1.13i·7-s − 2.44·9-s − 0.476·11-s + 0.137i·13-s + (−1.80 + 0.436i)15-s − 1.67i·17-s + 0.430·19-s − 2.10·21-s − 0.510i·23-s + (−0.889 + 0.457i)25-s − 2.68i·27-s − 0.721·29-s + 1.15·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.235 + 0.971i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.235 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.8427631325\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8427631325\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-13.1 - 54.3i)T \) |
good | 3 | \( 1 - 28.9iT - 243T^{2} \) |
| 7 | \( 1 - 146. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 191.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 83.9iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 2.00e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 677.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.29e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 3.26e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.15e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.13e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.05e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.29e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 9.52e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 1.47e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 3.82e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.58e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.17e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 5.13e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.33e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 1.59e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.33e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 5.13e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.08e4iT - 8.58e9T^{2} \) |
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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
−11.36888658169014918641714290056, −10.47509576234030495660559697822, −9.670471577851944699601101636517, −9.146216083190282372851735611590, −7.966138744420814012754638498487, −6.41920076102096641188422721543, −5.39975285190992694012480437009, −4.64622546463512280787081419634, −3.14794869640917152669270837185, −2.63699588327943709327990799062,
0.22833902311978010862296414529, 1.17828461587320731254994964167, 2.03968670472860891852103169549, 3.73676598956550360194402668012, 5.32840478218395409028792085268, 6.24026730283820534866658788907, 7.29398209034771539277155951564, 7.985140430034919595764272689683, 8.729073336372853418796554218307, 10.13178454727591890434916090370