| L(s) = 1 | + 81·3-s − 2.12e3·5-s + 1.44e3·7-s + 6.56e3·9-s − 2.01e4·11-s − 1.50e5·13-s − 1.72e5·15-s + 3.51e5·17-s + 5.84e5·19-s + 1.17e5·21-s + 2.89e5·23-s + 2.56e6·25-s + 5.31e5·27-s − 2.55e6·29-s + 9.90e6·31-s − 1.63e6·33-s − 3.07e6·35-s − 3.78e6·37-s − 1.21e7·39-s + 1.02e7·41-s + 3.60e6·43-s − 1.39e7·45-s + 1.06e7·47-s − 3.82e7·49-s + 2.84e7·51-s − 4.14e7·53-s + 4.28e7·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.52·5-s + 0.227·7-s + 0.333·9-s − 0.415·11-s − 1.45·13-s − 0.878·15-s + 1.02·17-s + 1.02·19-s + 0.131·21-s + 0.215·23-s + 1.31·25-s + 0.192·27-s − 0.670·29-s + 1.92·31-s − 0.239·33-s − 0.346·35-s − 0.332·37-s − 0.841·39-s + 0.567·41-s + 0.160·43-s − 0.507·45-s + 0.317·47-s − 0.948·49-s + 0.589·51-s − 0.720·53-s + 0.631·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - 81T \) |
| good | 5 | \( 1 + 2.12e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 1.44e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 2.01e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.50e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 3.51e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 5.84e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.89e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 2.55e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 9.90e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 3.78e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.02e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.60e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.06e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 4.14e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 9.14e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.51e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.97e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.32e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.50e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.92e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 4.76e7T + 1.86e17T^{2} \) |
| 89 | \( 1 + 1.00e9T + 3.50e17T^{2} \) |
| 97 | \( 1 - 8.13e8T + 7.60e17T^{2} \) |
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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
−9.370070361897159359473488883330, −8.034630893890571070283627042290, −7.80155238203800754666925695530, −6.93412025621487217493445464946, −5.27314610666690047500191808659, −4.44249137668299446081966450095, −3.39198146402209521981493590440, −2.60589625894087866184235501316, −1.06954436895772972613862566870, 0,
1.06954436895772972613862566870, 2.60589625894087866184235501316, 3.39198146402209521981493590440, 4.44249137668299446081966450095, 5.27314610666690047500191808659, 6.93412025621487217493445464946, 7.80155238203800754666925695530, 8.034630893890571070283627042290, 9.370070361897159359473488883330
