| L(s) = 1 | + 11.4·2-s + 2·4-s − 825·7-s − 1.43e3·8-s − 1.14e3·11-s + 5.07e3·13-s − 9.40e3·14-s − 1.66e4·16-s + 1.26e4·17-s − 1.20e4·19-s − 1.30e4·22-s − 1.54e4·23-s + 5.78e4·26-s − 1.65e3·28-s + 1.37e5·29-s + 8.87e4·31-s − 5.79e3·32-s + 1.44e5·34-s + 2.75e5·37-s − 1.37e5·38-s + 6.92e5·41-s − 1.34e5·43-s − 2.28e3·44-s − 1.76e5·46-s + 1.11e6·47-s − 1.42e5·49-s + 1.01e4·52-s + ⋯ |
| L(s) = 1 | + 1.00·2-s + 0.0156·4-s − 0.909·7-s − 0.992·8-s − 0.258·11-s + 0.640·13-s − 0.916·14-s − 1.01·16-s + 0.623·17-s − 0.402·19-s − 0.260·22-s − 0.264·23-s + 0.645·26-s − 0.0142·28-s + 1.05·29-s + 0.535·31-s − 0.0312·32-s + 0.628·34-s + 0.894·37-s − 0.405·38-s + 1.56·41-s − 0.257·43-s − 0.00403·44-s − 0.267·46-s + 1.56·47-s − 0.173·49-s + 0.0100·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.474944086\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.474944086\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 11.4T + 128T^{2} \) |
| 7 | \( 1 + 825T + 8.23e5T^{2} \) |
| 11 | \( 1 + 1.14e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 5.07e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.26e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.20e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.54e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.37e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 8.87e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.75e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 6.92e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 1.34e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.11e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.81e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 8.36e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.34e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.69e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 5.47e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.50e5T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.37e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 1.48e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 2.46e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.49e7T + 8.07e13T^{2} \) |
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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
−11.16253965092709106671091395653, −9.996076969509534313362058381445, −9.113085109656531284428357255570, −7.984103786013249628029125953405, −6.49905405483023117204101787474, −5.84789820040647607303517327302, −4.61488379285335764240482752810, −3.59968732673077525697614070131, −2.64282652279304575716129771023, −0.69362132328392822078269818979,
0.69362132328392822078269818979, 2.64282652279304575716129771023, 3.59968732673077525697614070131, 4.61488379285335764240482752810, 5.84789820040647607303517327302, 6.49905405483023117204101787474, 7.984103786013249628029125953405, 9.113085109656531284428357255570, 9.996076969509534313362058381445, 11.16253965092709106671091395653
