L(s) = 1 | − i·2-s − 4-s + i·7-s + i·8-s − 2i·13-s + 14-s + 16-s + 6i·17-s + 4·19-s − 2·26-s − i·28-s − 6·29-s − 4·31-s − i·32-s + 6·34-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 0.377i·7-s + 0.353i·8-s − 0.554i·13-s + 0.267·14-s + 0.250·16-s + 1.45i·17-s + 0.917·19-s − 0.392·26-s − 0.188i·28-s − 1.11·29-s − 0.718·31-s − 0.176i·32-s + 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.027850978\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.027850978\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 - 12iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 14iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.044553428327962660422903661411, −8.070456137943154175087844763333, −7.58573702554331648559733286775, −6.40995110952086286475092847198, −5.65829811136930735844716745151, −4.97876699452255437990135054285, −3.87879780623905568896066207502, −3.27108984758456766340236910427, −2.19972926975243127497532059098, −1.25579902994169803297794834921,
0.33278031377724664712408740075, 1.73758225492426653329847143699, 3.08532429289423874844650757683, 3.91047004660196640524981003685, 4.90978362149398567792015996993, 5.41163992768336156148376878162, 6.43265602712113626588566435157, 7.15401078043947886511583328200, 7.58218476486543759997009174505, 8.475599330455527378765883078711