| L(s) = 1 | − 4·7-s + 4·25-s − 32·37-s − 8·43-s − 2·49-s + 32·67-s − 16·79-s + 16·109-s + 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + 173-s − 16·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 4/5·25-s − 5.26·37-s − 1.21·43-s − 2/7·49-s + 3.90·67-s − 1.80·79-s + 1.53·109-s + 8/11·121-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 + 0.0783·163-s + 0.0773·167-s + 4/13·169-s + 0.0760·173-s − 1.20·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8180684839\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8180684839\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 28 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 - 124 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 118 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 170 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−8.668363417051754932643940315032, −8.508950907087926485732537358984, −8.444759433140861181874874284930, −8.020574892892539238019346912852, −7.911000192099818363452667037952, −7.27696966240240158037378687916, −6.94470607510634710894115864744, −6.90068257714039438901093338788, −6.77708867251991037798051935080, −6.68390668405292913129683867863, −5.96827714055916090833109149763, −5.91470088275298681401257152514, −5.54183921971613717507141750352, −5.15578607713144393009032285439, −4.88760039403958126294007338022, −4.87004105115356230309308212383, −4.04238038388355147148163060953, −3.96386430145868860875921572462, −3.34581103770000706355534884433, −3.23380941280560733707515626758, −3.19691365705484432783334280384, −2.43217406373457657783319091889, −1.88303140108365198059522200632, −1.61908072663052854791980645141, −0.46606084626972137445853824375,
0.46606084626972137445853824375, 1.61908072663052854791980645141, 1.88303140108365198059522200632, 2.43217406373457657783319091889, 3.19691365705484432783334280384, 3.23380941280560733707515626758, 3.34581103770000706355534884433, 3.96386430145868860875921572462, 4.04238038388355147148163060953, 4.87004105115356230309308212383, 4.88760039403958126294007338022, 5.15578607713144393009032285439, 5.54183921971613717507141750352, 5.91470088275298681401257152514, 5.96827714055916090833109149763, 6.68390668405292913129683867863, 6.77708867251991037798051935080, 6.90068257714039438901093338788, 6.94470607510634710894115864744, 7.27696966240240158037378687916, 7.911000192099818363452667037952, 8.020574892892539238019346912852, 8.444759433140861181874874284930, 8.508950907087926485732537358984, 8.668363417051754932643940315032
