| L(s) = 1 | + 9·3-s + 94·5-s − 144·7-s + 81·9-s + 380·11-s + 814·13-s + 846·15-s − 862·17-s + 1.15e3·19-s − 1.29e3·21-s + 488·23-s + 5.71e3·25-s + 729·27-s − 5.46e3·29-s − 9.56e3·31-s + 3.42e3·33-s − 1.35e4·35-s − 1.05e4·37-s + 7.32e3·39-s − 5.19e3·41-s + 1.70e4·43-s + 7.61e3·45-s − 3.16e3·47-s + 3.92e3·49-s − 7.75e3·51-s − 2.47e4·53-s + 3.57e4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.68·5-s − 1.11·7-s + 1/3·9-s + 0.946·11-s + 1.33·13-s + 0.970·15-s − 0.723·17-s + 0.734·19-s − 0.641·21-s + 0.192·23-s + 1.82·25-s + 0.192·27-s − 1.20·29-s − 1.78·31-s + 0.546·33-s − 1.86·35-s − 1.26·37-s + 0.771·39-s − 0.482·41-s + 1.40·43-s + 0.560·45-s − 0.209·47-s + 0.233·49-s − 0.417·51-s − 1.21·53-s + 1.59·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.417423290\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.417423290\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{2} T \) |
| good | 5 | \( 1 - 94 T + p^{5} T^{2} \) |
| 7 | \( 1 + 144 T + p^{5} T^{2} \) |
| 11 | \( 1 - 380 T + p^{5} T^{2} \) |
| 13 | \( 1 - 814 T + p^{5} T^{2} \) |
| 17 | \( 1 + 862 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1156 T + p^{5} T^{2} \) |
| 23 | \( 1 - 488 T + p^{5} T^{2} \) |
| 29 | \( 1 + 5466 T + p^{5} T^{2} \) |
| 31 | \( 1 + 9560 T + p^{5} T^{2} \) |
| 37 | \( 1 + 10506 T + p^{5} T^{2} \) |
| 41 | \( 1 + 5190 T + p^{5} T^{2} \) |
| 43 | \( 1 - 17084 T + p^{5} T^{2} \) |
| 47 | \( 1 + 3168 T + p^{5} T^{2} \) |
| 53 | \( 1 + 24770 T + p^{5} T^{2} \) |
| 59 | \( 1 + 17380 T + p^{5} T^{2} \) |
| 61 | \( 1 - 4366 T + p^{5} T^{2} \) |
| 67 | \( 1 - 5284 T + p^{5} T^{2} \) |
| 71 | \( 1 + 8360 T + p^{5} T^{2} \) |
| 73 | \( 1 - 39466 T + p^{5} T^{2} \) |
| 79 | \( 1 + 42376 T + p^{5} T^{2} \) |
| 83 | \( 1 - 61828 T + p^{5} T^{2} \) |
| 89 | \( 1 + 63078 T + p^{5} T^{2} \) |
| 97 | \( 1 + 16318 T + p^{5} T^{2} \) |
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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
−14.35092015175620567284805616259, −13.52631646608503332600722845001, −12.80163274166109632870158501431, −10.86505584726635450992232092569, −9.490019757982799925365178518986, −9.037760775378435410744068235563, −6.80981983447741917484942380503, −5.79297536113843096342983653929, −3.46839690829566703242110113859, −1.69860145812755597226835475246,
1.69860145812755597226835475246, 3.46839690829566703242110113859, 5.79297536113843096342983653929, 6.80981983447741917484942380503, 9.037760775378435410744068235563, 9.490019757982799925365178518986, 10.86505584726635450992232092569, 12.80163274166109632870158501431, 13.52631646608503332600722845001, 14.35092015175620567284805616259
