| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.007752824\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.007752824\) |
| \(L(\frac{7}{2})\) |
|
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
−11.88088301806723794894121168887, −10.87728341031689449171753490798, −9.508495346340074945698762755105, −8.352965817858873024279024479245, −7.77670086300355262601074918069, −6.83012866334038710928935276660, −4.75208489106450016535375834134, −4.12903051412120069785661326429, −2.66170034158824165971784865772, −0.885292495599894918388584968728,
0.885292495599894918388584968728, 2.66170034158824165971784865772, 4.12903051412120069785661326429, 4.75208489106450016535375834134, 6.83012866334038710928935276660, 7.77670086300355262601074918069, 8.352965817858873024279024479245, 9.508495346340074945698762755105, 10.87728341031689449171753490798, 11.88088301806723794894121168887
