| L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s + 4·13-s + 5·16-s + 4·17-s − 8·19-s + 4·23-s + 8·26-s + 4·29-s − 8·31-s + 6·32-s + 8·34-s − 4·37-s − 16·38-s + 12·41-s − 8·43-s + 8·46-s + 8·47-s + 12·52-s + 12·53-s + 8·58-s + 8·59-s − 12·61-s − 16·62-s + 7·64-s + 16·67-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s + 1.10·13-s + 5/4·16-s + 0.970·17-s − 1.83·19-s + 0.834·23-s + 1.56·26-s + 0.742·29-s − 1.43·31-s + 1.06·32-s + 1.37·34-s − 0.657·37-s − 2.59·38-s + 1.87·41-s − 1.21·43-s + 1.17·46-s + 1.16·47-s + 1.66·52-s + 1.64·53-s + 1.05·58-s + 1.04·59-s − 1.53·61-s − 2.03·62-s + 7/8·64-s + 1.95·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.558318308\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.558318308\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 48 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 26 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 12 T + 94 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 118 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 128 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 12 T + 152 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 154 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 138 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 160 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.927698328760850481354312932010, −8.854904646512019885680717432001, −8.260994610730353670738771461189, −7.993628104634749638104467263975, −7.35577307783285726690172344549, −7.17628674278209051487346451917, −6.71352982921670967095350357854, −6.31475450720530471303113856378, −5.87899271979687036458120926247, −5.68805730434565079173099842072, −5.23708276580679400235613009186, −4.69220445963268926386245835756, −4.32725631917608889902117991695, −3.94736337617112802129572639890, −3.37939618012230527233218531033, −3.28636743830996252219875624305, −2.38452022263727889040219876873, −2.17227626850892447233460060514, −1.41479414346004745625551718933, −0.74752741983909619393551071916,
0.74752741983909619393551071916, 1.41479414346004745625551718933, 2.17227626850892447233460060514, 2.38452022263727889040219876873, 3.28636743830996252219875624305, 3.37939618012230527233218531033, 3.94736337617112802129572639890, 4.32725631917608889902117991695, 4.69220445963268926386245835756, 5.23708276580679400235613009186, 5.68805730434565079173099842072, 5.87899271979687036458120926247, 6.31475450720530471303113856378, 6.71352982921670967095350357854, 7.17628674278209051487346451917, 7.35577307783285726690172344549, 7.993628104634749638104467263975, 8.260994610730353670738771461189, 8.854904646512019885680717432001, 8.927698328760850481354312932010
