| L(s) = 1 | + 3·5-s + 3·7-s − 3·9-s − 3·11-s + 5·13-s − 9·23-s + 25-s + 3·29-s + 9·31-s + 9·35-s + 4·37-s − 9·41-s − 9·43-s − 9·45-s − 3·47-s − 49-s − 9·55-s + 3·59-s + 61-s − 9·63-s + 15·65-s − 15·67-s + 24·71-s − 4·73-s − 9·77-s + 15·79-s + 9·81-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 1.13·7-s − 9-s − 0.904·11-s + 1.38·13-s − 1.87·23-s + 1/5·25-s + 0.557·29-s + 1.61·31-s + 1.52·35-s + 0.657·37-s − 1.40·41-s − 1.37·43-s − 1.34·45-s − 0.437·47-s − 1/7·49-s − 1.21·55-s + 0.390·59-s + 0.128·61-s − 1.13·63-s + 1.86·65-s − 1.83·67-s + 2.84·71-s − 0.468·73-s − 1.02·77-s + 1.68·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.468962942\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.468962942\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 3 T + 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 9 T + 68 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 9 T + 70 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 15 T + 142 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 15 T + 154 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 15 T + 142 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.49078372668303855757759986490, −13.19624302813122707111145947699, −12.04276712794267890174764583776, −11.99739986298116728067896772607, −11.23134466751041772860622014141, −10.90212705694862630101407144930, −10.14460845756410930506594181964, −9.991315260040037650100089460191, −9.269350462714940698997241122082, −8.323173913648606180816209590042, −8.308151510695133744638530031863, −7.909440859469040367913132712076, −6.62300201738300098617533839487, −6.25801030061991551753669093673, −5.65254390586573754776229900189, −5.19156925697800759713470792836, −4.43707803328412669519358222968, −3.38446480808265858762667512549, −2.41336522341264974174290916616, −1.63805317734805943639502836416,
1.63805317734805943639502836416, 2.41336522341264974174290916616, 3.38446480808265858762667512549, 4.43707803328412669519358222968, 5.19156925697800759713470792836, 5.65254390586573754776229900189, 6.25801030061991551753669093673, 6.62300201738300098617533839487, 7.909440859469040367913132712076, 8.308151510695133744638530031863, 8.323173913648606180816209590042, 9.269350462714940698997241122082, 9.991315260040037650100089460191, 10.14460845756410930506594181964, 10.90212705694862630101407144930, 11.23134466751041772860622014141, 11.99739986298116728067896772607, 12.04276712794267890174764583776, 13.19624302813122707111145947699, 13.49078372668303855757759986490
