| L(s) = 1 | + 2·3-s + 6·7-s + 9-s − 8·13-s + 2·19-s + 12·21-s − 9·25-s − 4·27-s − 16·31-s − 20·37-s − 16·39-s + 14·43-s + 13·49-s + 4·57-s − 10·61-s + 6·63-s − 30·73-s − 18·75-s + 8·79-s − 11·81-s − 48·91-s − 32·93-s + 32·97-s + 28·103-s + 24·109-s − 40·111-s − 8·117-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 2.26·7-s + 1/3·9-s − 2.21·13-s + 0.458·19-s + 2.61·21-s − 9/5·25-s − 0.769·27-s − 2.87·31-s − 3.28·37-s − 2.56·39-s + 2.13·43-s + 13/7·49-s + 0.529·57-s − 1.28·61-s + 0.755·63-s − 3.51·73-s − 2.07·75-s + 0.900·79-s − 1.22·81-s − 5.03·91-s − 3.31·93-s + 3.24·97-s + 2.75·103-s + 2.29·109-s − 3.79·111-s − 0.739·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
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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
−7.79236899027639545070642498110, −7.53818989363645791606962986645, −7.41110136347227768123475878009, −7.20024501520191211813153160377, −6.01501469952133530659188791248, −5.62405161804576690819108289841, −5.15976497091815013069726027465, −4.78246098563165761782711476378, −4.35808362404597816678481770906, −3.62617557238284773485534023353, −3.25598629803065603741482539939, −2.33036552779137826129420876510, −1.81707434889353167746748104685, −1.81226351589066908089801551987, 0,
1.81226351589066908089801551987, 1.81707434889353167746748104685, 2.33036552779137826129420876510, 3.25598629803065603741482539939, 3.62617557238284773485534023353, 4.35808362404597816678481770906, 4.78246098563165761782711476378, 5.15976497091815013069726027465, 5.62405161804576690819108289841, 6.01501469952133530659188791248, 7.20024501520191211813153160377, 7.41110136347227768123475878009, 7.53818989363645791606962986645, 7.79236899027639545070642498110
