| L(s) = 1 | + 127.·3-s + 393.·5-s + 6.60e3·7-s − 3.54e3·9-s + 5.50e4·11-s + 2.85e4·13-s + 4.99e4·15-s − 2.29e5·17-s + 3.70e5·19-s + 8.38e5·21-s + 1.83e6·23-s − 1.79e6·25-s − 2.95e6·27-s + 1.20e6·29-s + 5.78e6·31-s + 6.99e6·33-s + 2.59e6·35-s − 8.39e5·37-s + 3.62e6·39-s + 2.52e6·41-s + 1.28e6·43-s − 1.39e6·45-s + 1.25e7·47-s + 3.22e6·49-s − 2.91e7·51-s − 9.21e7·53-s + 2.16e7·55-s + ⋯ |
| L(s) = 1 | + 0.905·3-s + 0.281·5-s + 1.03·7-s − 0.179·9-s + 1.13·11-s + 0.277·13-s + 0.254·15-s − 0.665·17-s + 0.651·19-s + 0.941·21-s + 1.36·23-s − 0.920·25-s − 1.06·27-s + 0.316·29-s + 1.12·31-s + 1.02·33-s + 0.292·35-s − 0.0736·37-s + 0.251·39-s + 0.139·41-s + 0.0573·43-s − 0.0506·45-s + 0.374·47-s + 0.0798·49-s − 0.603·51-s − 1.60·53-s + 0.319·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(3.761326800\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.761326800\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - 2.85e4T \) |
| good | 3 | \( 1 - 127.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 393.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 6.60e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 5.50e4T + 2.35e9T^{2} \) |
| 17 | \( 1 + 2.29e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 3.70e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.83e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 1.20e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 5.78e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 8.39e5T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.52e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.28e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.25e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 9.21e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.25e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.80e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.32e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.13e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 5.38e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.37e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 1.78e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 9.93e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.12e9T + 7.60e17T^{2} \) |
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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
−11.82876039667493926921111406302, −11.07450769888314845728010512486, −9.547498482413209862968684473478, −8.742577487346069470133342727179, −7.82703135862238798829577836711, −6.45985313920166278791027778353, −4.98142703010444177016060098856, −3.65275010992777513860588477744, −2.30165984431038222347682306819, −1.12472266825986265436372713806,
1.12472266825986265436372713806, 2.30165984431038222347682306819, 3.65275010992777513860588477744, 4.98142703010444177016060098856, 6.45985313920166278791027778353, 7.82703135862238798829577836711, 8.742577487346069470133342727179, 9.547498482413209862968684473478, 11.07450769888314845728010512486, 11.82876039667493926921111406302
