| L(s) = 1 | − 2·3-s + 2·5-s − 2·7-s + 9-s − 4·13-s − 4·15-s + 14·19-s + 4·21-s + 25-s + 2·27-s − 24·29-s + 2·31-s − 4·35-s + 2·37-s + 8·39-s − 4·43-s + 2·45-s + 12·47-s + 7·49-s − 28·57-s + 12·59-s + 8·61-s − 2·63-s − 8·65-s + 2·67-s − 10·73-s − 2·75-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s − 0.755·7-s + 1/3·9-s − 1.10·13-s − 1.03·15-s + 3.21·19-s + 0.872·21-s + 1/5·25-s + 0.384·27-s − 4.45·29-s + 0.359·31-s − 0.676·35-s + 0.328·37-s + 1.28·39-s − 0.609·43-s + 0.298·45-s + 1.75·47-s + 49-s − 3.70·57-s + 1.56·59-s + 1.02·61-s − 0.251·63-s − 0.992·65-s + 0.244·67-s − 1.17·73-s − 0.230·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7724779358\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7724779358\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| good | 11 | $C_2^3$ | \( 1 - 4 T^{2} - 105 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 2 T + 9 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 16 T^{2} - 33 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 28 T^{2} + 255 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 12 T + 76 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 2 T + 13 T^{2} + 142 T^{3} - 1004 T^{4} + 142 p T^{5} + 13 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 2 T - 53 T^{2} + 34 T^{3} + 1732 T^{4} + 34 p T^{5} - 53 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 + 64 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 2 T + 69 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^3$ | \( 1 - 34 T^{2} - 1653 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 12 T + 8 T^{2} - 216 T^{3} + 6519 T^{4} - 216 p T^{5} + 8 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 8 T - 2 T^{2} + 448 T^{3} - 3269 T^{4} + 448 p T^{5} - 2 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 2 T - 113 T^{2} + 34 T^{3} + 8932 T^{4} + 34 p T^{5} - 113 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 10 T - 53 T^{2} + 70 T^{3} + 10780 T^{4} + 70 p T^{5} - 53 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 + 12 T + 184 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 12 T - 52 T^{2} - 216 T^{3} + 17679 T^{4} - 216 p T^{5} - 52 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 16 T + 186 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.83347337636037704019412163322, −7.65034272377223948055316268734, −7.62314133310564403429009010386, −7.27016364787680368618856649709, −7.17193146483779123811676426189, −7.01852940299237952409717322203, −6.55111335422242991701834561640, −6.30159179171425474207427120701, −5.88716998267454923762085605991, −5.74860372733934330100202842154, −5.66799939797787732853098303803, −5.31768711208281404112084047763, −5.27795176955433360983457336322, −4.94357592168906617971121031521, −4.58061929628997007073982246359, −4.16337751868363385412291349195, −3.66946469448207404319527550891, −3.55013538215165469220858220608, −3.44622082449789052499283315338, −2.68503386982822601627751115173, −2.38530391776947463496428850849, −2.37443482071411292032872693957, −1.36465581120633051088249575763, −1.35276220967991546484170318303, −0.37023026769268605719783216471,
0.37023026769268605719783216471, 1.35276220967991546484170318303, 1.36465581120633051088249575763, 2.37443482071411292032872693957, 2.38530391776947463496428850849, 2.68503386982822601627751115173, 3.44622082449789052499283315338, 3.55013538215165469220858220608, 3.66946469448207404319527550891, 4.16337751868363385412291349195, 4.58061929628997007073982246359, 4.94357592168906617971121031521, 5.27795176955433360983457336322, 5.31768711208281404112084047763, 5.66799939797787732853098303803, 5.74860372733934330100202842154, 5.88716998267454923762085605991, 6.30159179171425474207427120701, 6.55111335422242991701834561640, 7.01852940299237952409717322203, 7.17193146483779123811676426189, 7.27016364787680368618856649709, 7.62314133310564403429009010386, 7.65034272377223948055316268734, 7.83347337636037704019412163322
