| L(s) = 1 | + 28.7·3-s + 25·5-s + 42.1·7-s + 581.·9-s − 416.·11-s + 966.·13-s + 717.·15-s − 1.83e3·17-s − 317.·19-s + 1.21e3·21-s − 1.56e3·23-s + 625·25-s + 9.72e3·27-s + 7.75e3·29-s − 102.·31-s − 1.19e4·33-s + 1.05e3·35-s + 1.93e3·37-s + 2.77e4·39-s + 7.99e3·41-s − 1.65e4·43-s + 1.45e4·45-s − 1.86e4·47-s − 1.50e4·49-s − 5.26e4·51-s − 1.49e4·53-s − 1.04e4·55-s + ⋯ |
| L(s) = 1 | + 1.84·3-s + 0.447·5-s + 0.325·7-s + 2.39·9-s − 1.03·11-s + 1.58·13-s + 0.823·15-s − 1.53·17-s − 0.201·19-s + 0.598·21-s − 0.618·23-s + 0.200·25-s + 2.56·27-s + 1.71·29-s − 0.0191·31-s − 1.91·33-s + 0.145·35-s + 0.232·37-s + 2.92·39-s + 0.742·41-s − 1.36·43-s + 1.07·45-s − 1.23·47-s − 0.894·49-s − 2.83·51-s − 0.732·53-s − 0.463·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.541351928\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.541351928\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 25T \) |
| good | 3 | \( 1 - 28.7T + 243T^{2} \) |
| 7 | \( 1 - 42.1T + 1.68e4T^{2} \) |
| 11 | \( 1 + 416.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 966.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.83e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 317.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.56e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.75e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 102.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.93e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.99e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.65e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.86e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.49e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.98e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.80e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.50e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.12e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.01e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.40e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.05e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.07e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.11e4T + 8.58e9T^{2} \) |
| 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
−13.52264286000985000523930192144, −12.92523797441355372694644036642, −10.95955500367018888610578644902, −9.843976300791482150622432771857, −8.622647609257301174494213539937, −8.119619806704461866264543309587, −6.54488774356521669943509936710, −4.47386861212425605095010252040, −3.00116954620236432032596373564, −1.76865176153048056221471300631,
1.76865176153048056221471300631, 3.00116954620236432032596373564, 4.47386861212425605095010252040, 6.54488774356521669943509936710, 8.119619806704461866264543309587, 8.622647609257301174494213539937, 9.843976300791482150622432771857, 10.95955500367018888610578644902, 12.92523797441355372694644036642, 13.52264286000985000523930192144
