| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s + 3·11-s + 4·13-s − 14-s + 16-s + 2·19-s − 3·22-s − 6·23-s − 4·26-s + 28-s − 6·29-s + 5·31-s − 32-s − 2·37-s − 2·38-s + 6·41-s + 10·43-s + 3·44-s + 6·46-s + 6·47-s − 6·49-s + 4·52-s + 9·53-s − 56-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s + 0.904·11-s + 1.10·13-s − 0.267·14-s + 1/4·16-s + 0.458·19-s − 0.639·22-s − 1.25·23-s − 0.784·26-s + 0.188·28-s − 1.11·29-s + 0.898·31-s − 0.176·32-s − 0.328·37-s − 0.324·38-s + 0.937·41-s + 1.52·43-s + 0.452·44-s + 0.884·46-s + 0.875·47-s − 6/7·49-s + 0.554·52-s + 1.23·53-s − 0.133·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.382590144\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.382590144\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−9.424555904372419494656746065633, −8.904247934298067766833725269707, −8.033154449173421852738549606568, −7.36566276940036406451155206076, −6.29951651895743522476705768290, −5.74395141102766134561119794703, −4.34568268458938072273214663423, −3.49841666048700785356624668043, −2.10144548480492536578842252757, −1.01053650029556092883839412457,
1.01053650029556092883839412457, 2.10144548480492536578842252757, 3.49841666048700785356624668043, 4.34568268458938072273214663423, 5.74395141102766134561119794703, 6.29951651895743522476705768290, 7.36566276940036406451155206076, 8.033154449173421852738549606568, 8.904247934298067766833725269707, 9.424555904372419494656746065633
