| L(s) = 1 | + 2·5-s − 6·9-s − 8·19-s + 5·25-s − 8·29-s − 16·31-s − 24·41-s − 12·45-s + 8·59-s − 12·61-s + 48·71-s − 8·79-s + 9·81-s + 20·89-s − 16·95-s − 36·101-s + 28·109-s + 22·121-s + 22·125-s + 127-s + 131-s + 137-s + 139-s − 16·145-s + 149-s + 151-s − 32·155-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 2·9-s − 1.83·19-s + 25-s − 1.48·29-s − 2.87·31-s − 3.74·41-s − 1.78·45-s + 1.04·59-s − 1.53·61-s + 5.69·71-s − 0.900·79-s + 81-s + 2.11·89-s − 1.64·95-s − 3.58·101-s + 2.68·109-s + 2·121-s + 1.96·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.32·145-s + 0.0819·149-s + 0.0813·151-s − 2.57·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5303379889\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5303379889\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^3$ | \( 1 + 18 T^{2} + 35 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 18 T^{2} - 205 T^{4} - 18 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 10 T^{2} - 1269 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 30 T^{2} - 1309 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 70 T^{2} + 411 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 + 130 T^{2} + 11571 T^{4} + 130 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2}( 1 + 17 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 - 10 T + 11 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−7.06778963287421324345818032641, −6.84868433774256190098834897166, −6.68566993326071901583519725304, −6.45697787158082993671958087882, −6.36583677071182410758661826612, −5.89640853563513115239640527874, −5.85801455012162503040410374454, −5.50618360276221769350501894068, −5.34148652099241314468368917245, −5.20129556572070290521420751281, −5.05341900623645401577263226458, −4.73781527171517681112547813408, −4.31342700986962301769317605282, −3.96993707716611406404570655210, −3.85946822758084373859720654583, −3.32869870830772242983739240042, −3.28308140919815149127699104725, −3.25511310276011531471851352211, −2.52183853143902469190831295643, −2.41481317178171931570428039069, −1.89919337600174712664439897780, −1.87738825784685315597162427219, −1.70214572125373996899278645018, −0.76086699125454369558400739701, −0.19365468955560769650862945352,
0.19365468955560769650862945352, 0.76086699125454369558400739701, 1.70214572125373996899278645018, 1.87738825784685315597162427219, 1.89919337600174712664439897780, 2.41481317178171931570428039069, 2.52183853143902469190831295643, 3.25511310276011531471851352211, 3.28308140919815149127699104725, 3.32869870830772242983739240042, 3.85946822758084373859720654583, 3.96993707716611406404570655210, 4.31342700986962301769317605282, 4.73781527171517681112547813408, 5.05341900623645401577263226458, 5.20129556572070290521420751281, 5.34148652099241314468368917245, 5.50618360276221769350501894068, 5.85801455012162503040410374454, 5.89640853563513115239640527874, 6.36583677071182410758661826612, 6.45697787158082993671958087882, 6.68566993326071901583519725304, 6.84868433774256190098834897166, 7.06778963287421324345818032641
