| L(s) = 1 | − 2·7-s + 3·11-s + 2·13-s + 6·17-s + 2·19-s − 6·23-s + 5·25-s − 6·29-s + 4·31-s + 8·37-s + 9·41-s − 43-s − 6·47-s + 7·49-s + 24·53-s − 3·59-s + 8·61-s + 5·67-s + 24·71-s + 22·73-s − 6·77-s + 4·79-s − 12·83-s − 12·89-s − 4·91-s − 5·97-s − 14·103-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 0.904·11-s + 0.554·13-s + 1.45·17-s + 0.458·19-s − 1.25·23-s + 25-s − 1.11·29-s + 0.718·31-s + 1.31·37-s + 1.40·41-s − 0.152·43-s − 0.875·47-s + 49-s + 3.29·53-s − 0.390·59-s + 1.02·61-s + 0.610·67-s + 2.84·71-s + 2.57·73-s − 0.683·77-s + 0.450·79-s − 1.31·83-s − 1.27·89-s − 0.419·91-s − 0.507·97-s − 1.37·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2985984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2985984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.941423431\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.941423431\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−9.549806371349108868203495748201, −9.319306913647557597465532222421, −8.578299566267676138784058443742, −8.444374321850559382581779737883, −7.983769894164462985191378108304, −7.51617003450004046951868801994, −7.10328952546716103278246925950, −6.75738936526882024853228527373, −6.22699178373841196112329328148, −5.99383077044232126641723392891, −5.39729599067915389895786572042, −5.29582230259955406923561751020, −4.34655751342043486791042279752, −4.06474378806698340799811013098, −3.52782716161226910439648563119, −3.35831520208979988418623423211, −2.45644163383220243230415057731, −2.17232066437258042670374245009, −1.01084605459179801569980299113, −0.859199978710137924316585558396,
0.859199978710137924316585558396, 1.01084605459179801569980299113, 2.17232066437258042670374245009, 2.45644163383220243230415057731, 3.35831520208979988418623423211, 3.52782716161226910439648563119, 4.06474378806698340799811013098, 4.34655751342043486791042279752, 5.29582230259955406923561751020, 5.39729599067915389895786572042, 5.99383077044232126641723392891, 6.22699178373841196112329328148, 6.75738936526882024853228527373, 7.10328952546716103278246925950, 7.51617003450004046951868801994, 7.983769894164462985191378108304, 8.444374321850559382581779737883, 8.578299566267676138784058443742, 9.319306913647557597465532222421, 9.549806371349108868203495748201
