| L(s) = 1 | + 231.·3-s + 1.36e3·5-s − 3.00e3·7-s + 3.40e4·9-s − 8.49e4·11-s − 2.85e4·13-s + 3.15e5·15-s − 3.58e5·17-s − 3.15e5·19-s − 6.96e5·21-s − 6.49e5·23-s − 9.48e4·25-s + 3.31e6·27-s − 4.28e6·29-s − 1.18e6·31-s − 1.96e7·33-s − 4.09e6·35-s + 5.51e6·37-s − 6.61e6·39-s − 2.57e5·41-s + 7.99e5·43-s + 4.63e7·45-s − 3.91e7·47-s − 3.13e7·49-s − 8.29e7·51-s + 3.04e7·53-s − 1.15e8·55-s + ⋯ |
| L(s) = 1 | + 1.65·3-s + 0.975·5-s − 0.473·7-s + 1.72·9-s − 1.75·11-s − 0.277·13-s + 1.61·15-s − 1.04·17-s − 0.554·19-s − 0.781·21-s − 0.484·23-s − 0.0485·25-s + 1.20·27-s − 1.12·29-s − 0.231·31-s − 2.89·33-s − 0.461·35-s + 0.483·37-s − 0.458·39-s − 0.0142·41-s + 0.0356·43-s + 1.68·45-s − 1.17·47-s − 0.776·49-s − 1.71·51-s + 0.530·53-s − 1.70·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 - 231.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.36e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 3.00e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 8.49e4T + 2.35e9T^{2} \) |
| 17 | \( 1 + 3.58e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 3.15e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 6.49e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 4.28e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 1.18e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 5.51e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.57e5T + 3.27e14T^{2} \) |
| 43 | \( 1 - 7.99e5T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.91e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 3.04e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.67e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 6.85e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.58e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.12e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.19e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.88e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 6.75e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 1.13e9T + 3.50e17T^{2} \) |
| 97 | \( 1 + 3.65e8T + 7.60e17T^{2} \) |
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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
−9.914133985360739554774884910086, −9.338374475747748933542889992097, −8.305276858613281929013537947262, −7.51883525957041780002949349111, −6.24413637483546910943741990348, −4.93080535678196584369151897101, −3.54227124432324091289546281913, −2.42393077355455098959891392874, −2.00410422293429755484992919432, 0,
2.00410422293429755484992919432, 2.42393077355455098959891392874, 3.54227124432324091289546281913, 4.93080535678196584369151897101, 6.24413637483546910943741990348, 7.51883525957041780002949349111, 8.305276858613281929013537947262, 9.338374475747748933542889992097, 9.914133985360739554774884910086
