| L(s) = 1 | + 4-s + 9-s − 6·11-s + 10·19-s + 20·31-s + 36-s + 36·41-s − 6·44-s − 2·49-s + 24·59-s − 16·61-s − 64-s − 24·71-s + 10·76-s + 16·79-s + 12·89-s − 6·99-s + 28·109-s + 31·121-s + 20·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1/3·9-s − 1.80·11-s + 2.29·19-s + 3.59·31-s + 1/6·36-s + 5.62·41-s − 0.904·44-s − 2/7·49-s + 3.12·59-s − 2.04·61-s − 1/8·64-s − 2.84·71-s + 1.14·76-s + 1.80·79-s + 1.27·89-s − 0.603·99-s + 2.68·109-s + 2.81·121-s + 1.79·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.132075190\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.132075190\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| good | 11 | $C_2^2$ | \( ( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 35 T^{2} + 696 T^{4} - 35 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 10 T + 69 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$$\times$$C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )( 1 + 47 T^{2} + p^{2} T^{4} ) \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 85 T^{2} + 5016 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 97 T^{2} + 6600 T^{4} + 97 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 70 T^{2} + 411 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 + 142 T^{2} + 14835 T^{4} + 142 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \) |
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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
−7.30062456997015354675758620118, −6.83668335869530616314717622520, −6.71642180873984443793157184802, −6.29969178161775169190159161496, −6.26819688794257191531802327220, −5.86070959764679292981238071897, −5.72004099072674477372166456788, −5.66446051003091680172721572848, −5.44052442387589575266200077633, −4.74632560401119445191610043849, −4.73176338026048808817195068439, −4.65900193572421154527749653667, −4.62854885301692593513860835370, −3.99138408093282482985659675388, −3.76504292512899660935286053061, −3.48043691848987554180668374927, −3.18760925455184397667499783025, −2.68864772920615820044301548468, −2.67691697405132522064789992797, −2.47701985593217600438889990559, −2.37482594759004623031554271283, −1.63504862714991164852672097446, −1.22134477483076910399290262150, −0.74027399167871410108097694722, −0.73403157336136968534609208242,
0.73403157336136968534609208242, 0.74027399167871410108097694722, 1.22134477483076910399290262150, 1.63504862714991164852672097446, 2.37482594759004623031554271283, 2.47701985593217600438889990559, 2.67691697405132522064789992797, 2.68864772920615820044301548468, 3.18760925455184397667499783025, 3.48043691848987554180668374927, 3.76504292512899660935286053061, 3.99138408093282482985659675388, 4.62854885301692593513860835370, 4.65900193572421154527749653667, 4.73176338026048808817195068439, 4.74632560401119445191610043849, 5.44052442387589575266200077633, 5.66446051003091680172721572848, 5.72004099072674477372166456788, 5.86070959764679292981238071897, 6.26819688794257191531802327220, 6.29969178161775169190159161496, 6.71642180873984443793157184802, 6.83668335869530616314717622520, 7.30062456997015354675758620118
