| L(s) = 1 | + 3·3-s + 5·7-s + 6·9-s − 10·13-s + 15·19-s + 15·21-s − 5·25-s + 9·27-s − 3·31-s − 11·37-s − 30·39-s + 18·49-s + 45·57-s − 14·61-s + 30·63-s − 27·67-s + 17·73-s − 15·75-s − 9·79-s + 9·81-s − 50·91-s − 9·93-s − 28·97-s + 27·103-s − 19·109-s − 33·111-s − 60·117-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 1.88·7-s + 2·9-s − 2.77·13-s + 3.44·19-s + 3.27·21-s − 25-s + 1.73·27-s − 0.538·31-s − 1.80·37-s − 4.80·39-s + 18/7·49-s + 5.96·57-s − 1.79·61-s + 3.77·63-s − 3.29·67-s + 1.98·73-s − 1.73·75-s − 1.01·79-s + 81-s − 5.24·91-s − 0.933·93-s − 2.84·97-s + 2.66·103-s − 1.81·109-s − 3.13·111-s − 5.54·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.364377441\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.364377441\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−11.84787519526608028761271089747, −11.57472253825768575479837927905, −10.69112174197364719611145967863, −10.30435249875699025265857053051, −9.631557988697083145006011728914, −9.529110637228711230302714790753, −9.073966290637435842939870337236, −8.445606080014714772918313551743, −7.76416354164532518642408849618, −7.68654730571174364996852594099, −7.32084954778031383962977395575, −7.01632231814056980011308061146, −5.47217738869203105588358474387, −5.37129975544239431944124468336, −4.69614039300713849480165679118, −4.26904017190803662895352978915, −3.24880393759034102009435323129, −2.90000554987988409497904051549, −2.01763161693081070318614651528, −1.51858851308555651634883171078,
1.51858851308555651634883171078, 2.01763161693081070318614651528, 2.90000554987988409497904051549, 3.24880393759034102009435323129, 4.26904017190803662895352978915, 4.69614039300713849480165679118, 5.37129975544239431944124468336, 5.47217738869203105588358474387, 7.01632231814056980011308061146, 7.32084954778031383962977395575, 7.68654730571174364996852594099, 7.76416354164532518642408849618, 8.445606080014714772918313551743, 9.073966290637435842939870337236, 9.529110637228711230302714790753, 9.631557988697083145006011728914, 10.30435249875699025265857053051, 10.69112174197364719611145967863, 11.57472253825768575479837927905, 11.84787519526608028761271089747
