| L(s) = 1 | + 81·3-s − 397.·5-s − 7.34e3·7-s + 6.56e3·9-s − 5.53e4·11-s − 3.40e3·13-s − 3.22e4·15-s − 1.55e5·17-s + 2.05e5·19-s − 5.94e5·21-s − 1.58e5·23-s − 1.79e6·25-s + 5.31e5·27-s − 1.82e6·29-s + 1.92e6·31-s − 4.48e6·33-s + 2.91e6·35-s − 4.17e6·37-s − 2.76e5·39-s − 1.70e7·41-s − 3.26e7·43-s − 2.60e6·45-s + 3.89e7·47-s + 1.35e7·49-s − 1.25e7·51-s + 4.64e7·53-s + 2.20e7·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.284·5-s − 1.15·7-s + 0.333·9-s − 1.14·11-s − 0.0331·13-s − 0.164·15-s − 0.450·17-s + 0.361·19-s − 0.667·21-s − 0.118·23-s − 0.919·25-s + 0.192·27-s − 0.478·29-s + 0.373·31-s − 0.658·33-s + 0.328·35-s − 0.366·37-s − 0.0191·39-s − 0.941·41-s − 1.45·43-s − 0.0948·45-s + 1.16·47-s + 0.335·49-s − 0.260·51-s + 0.808·53-s + 0.324·55-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
| \(L(5)\) |
\(\approx\) |
\(1.108313895\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.108313895\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 + 397.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 7.34e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 5.53e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 3.40e3T + 1.06e10T^{2} \) |
| 17 | \( 1 + 1.55e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 2.05e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.58e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.82e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 1.92e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 4.17e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.70e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.26e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.89e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 4.64e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.03e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 4.66e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 6.63e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 6.28e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.04e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.63e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 5.37e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 2.14e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.21e9T + 7.60e17T^{2} \) |
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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
−9.838815731298791423596047001332, −8.915608575732974893400844479179, −7.965777190736857954528717690221, −7.15131417470212469162272865647, −6.13331777737793219390281676115, −5.00064938426459951183435895809, −3.75492545150283330551566731300, −2.98467667422709821432920041060, −1.97987009520064533181380419620, −0.40407642548045231245487673679,
0.40407642548045231245487673679, 1.97987009520064533181380419620, 2.98467667422709821432920041060, 3.75492545150283330551566731300, 5.00064938426459951183435895809, 6.13331777737793219390281676115, 7.15131417470212469162272865647, 7.965777190736857954528717690221, 8.915608575732974893400844479179, 9.838815731298791423596047001332
