| L(s) = 1 | − 2·3-s + 3.31·5-s − 3.31·7-s + 9-s + 5·11-s − 6.63·15-s + 5·17-s + 19-s + 6.63·21-s − 6.63·23-s + 6·25-s + 4·27-s + 6.63·29-s − 10·33-s − 11·35-s − 6.63·37-s + 6·41-s − 43-s + 3.31·45-s + 9.94·47-s + 4·49-s − 10·51-s − 13.2·53-s + 16.5·55-s − 2·57-s − 6·59-s + 9.94·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.48·5-s − 1.25·7-s + 0.333·9-s + 1.50·11-s − 1.71·15-s + 1.21·17-s + 0.229·19-s + 1.44·21-s − 1.38·23-s + 1.20·25-s + 0.769·27-s + 1.23·29-s − 1.74·33-s − 1.85·35-s − 1.09·37-s + 0.937·41-s − 0.152·43-s + 0.494·45-s + 1.45·47-s + 0.571·49-s − 1.40·51-s − 1.82·53-s + 2.23·55-s − 0.264·57-s − 0.781·59-s + 1.27·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.579185465\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.579185465\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2T + 3T^{2} \) |
| 5 | \( 1 - 3.31T + 5T^{2} \) |
| 7 | \( 1 + 3.31T + 7T^{2} \) |
| 11 | \( 1 - 5T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 5T + 17T^{2} \) |
| 23 | \( 1 + 6.63T + 23T^{2} \) |
| 29 | \( 1 - 6.63T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 6.63T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 - 9.94T + 47T^{2} \) |
| 53 | \( 1 + 13.2T + 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 - 9.94T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 - 6.63T + 71T^{2} \) |
| 73 | \( 1 - 9T + 73T^{2} \) |
| 79 | \( 1 + 13.2T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - 4T + 89T^{2} \) |
| 97 | \( 1 + 12T + 97T^{2} \) |
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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
−8.408955934204590632298213978341, −7.17772070487467020282965437766, −6.48121400310534845326283856079, −6.04532390316239214897641829065, −5.73048567047010105943877079604, −4.77916638273426597161291059640, −3.73731650126429940795228045054, −2.87847179444535643173325927741, −1.69915756574590723725870940623, −0.76117520015341031534849437195,
0.76117520015341031534849437195, 1.69915756574590723725870940623, 2.87847179444535643173325927741, 3.73731650126429940795228045054, 4.77916638273426597161291059640, 5.73048567047010105943877079604, 6.04532390316239214897641829065, 6.48121400310534845326283856079, 7.17772070487467020282965437766, 8.408955934204590632298213978341
