| L(s) = 1 | + 2·3-s + 2·5-s − 2·7-s + 3·9-s + 4·15-s − 8·17-s − 8·19-s − 4·21-s + 10·23-s + 25-s + 10·27-s − 6·29-s − 12·31-s − 4·35-s − 24·37-s + 10·41-s − 4·43-s + 6·45-s + 14·47-s + 9·49-s − 16·51-s − 8·53-s − 16·57-s + 12·59-s + 6·61-s − 6·63-s + 2·67-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s − 0.755·7-s + 9-s + 1.03·15-s − 1.94·17-s − 1.83·19-s − 0.872·21-s + 2.08·23-s + 1/5·25-s + 1.92·27-s − 1.11·29-s − 2.15·31-s − 0.676·35-s − 3.94·37-s + 1.56·41-s − 0.609·43-s + 0.894·45-s + 2.04·47-s + 9/7·49-s − 2.24·51-s − 1.09·53-s − 2.11·57-s + 1.56·59-s + 0.768·61-s − 0.755·63-s + 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.483836693\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.483836693\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| good | 7 | $D_4\times C_2$ | \( 1 + 2 T - 5 T^{2} - 10 T^{3} + 4 T^{4} - 10 p T^{5} - 5 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 19 | $D_{4}$ | \( ( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 10 T + 35 T^{2} - 190 T^{3} + 1396 T^{4} - 190 p T^{5} + 35 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 6 T - 7 T^{2} - 90 T^{3} - 36 T^{4} - 90 p T^{5} - 7 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 12 T + 70 T^{2} + 144 T^{3} + 51 T^{4} + 144 p T^{5} + 70 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 41 | $D_4\times C_2$ | \( 1 - 10 T + 17 T^{2} - 10 T^{3} + 1108 T^{4} - 10 p T^{5} + 17 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 4 T + 22 T^{2} - 368 T^{3} - 2501 T^{4} - 368 p T^{5} + 22 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 14 T + 59 T^{2} - 602 T^{3} + 7348 T^{4} - 602 p T^{5} + 59 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 12 T + 14 T^{2} - 144 T^{3} + 4923 T^{4} - 144 p T^{5} + 14 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 3 T - 52 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 2 T + 19 T^{2} + 298 T^{3} - 4532 T^{4} + 298 p T^{5} + 19 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 4 T - 122 T^{2} + 80 T^{3} + 11539 T^{4} + 80 p T^{5} - 122 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 6 T - 133 T^{2} - 18 T^{3} + 18684 T^{4} - 18 p T^{5} - 133 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 14 T + 131 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.410216165895431156306392478615, −8.213947696406722007585818302209, −7.81992017963868845293862917471, −7.58033622860044370313005924892, −7.07107515254452554170514653658, −6.92719753273848474696197163068, −6.80474761404584793533608593350, −6.73509164905450659471738177886, −6.56996510580662521243563312487, −5.99258756844737649999135968223, −5.56119023600477559060575936366, −5.46436336555606318863860802678, −5.21943140144790973909746308195, −5.03210393698369414987593930865, −4.31804690430714939877816087568, −4.21677105280676442361984230932, −3.99071748656861140630327823532, −3.70179882711711695871574156615, −3.18251269739172063177625062524, −2.98189148802267898869008997397, −2.50688123047966187989500561819, −2.26102702955856286519722017493, −1.80144596751132884212957624817, −1.69963129427529424521091621478, −0.55773988127577540184293511907,
0.55773988127577540184293511907, 1.69963129427529424521091621478, 1.80144596751132884212957624817, 2.26102702955856286519722017493, 2.50688123047966187989500561819, 2.98189148802267898869008997397, 3.18251269739172063177625062524, 3.70179882711711695871574156615, 3.99071748656861140630327823532, 4.21677105280676442361984230932, 4.31804690430714939877816087568, 5.03210393698369414987593930865, 5.21943140144790973909746308195, 5.46436336555606318863860802678, 5.56119023600477559060575936366, 5.99258756844737649999135968223, 6.56996510580662521243563312487, 6.73509164905450659471738177886, 6.80474761404584793533608593350, 6.92719753273848474696197163068, 7.07107515254452554170514653658, 7.58033622860044370313005924892, 7.81992017963868845293862917471, 8.213947696406722007585818302209, 8.410216165895431156306392478615
