| L(s) = 1 | − 3-s − 2·4-s + 9-s + 6·11-s + 2·12-s − 2·13-s + 4·16-s − 18·23-s − 25-s − 27-s − 6·33-s − 2·36-s − 8·37-s + 2·39-s − 12·44-s + 12·47-s − 4·48-s + 2·49-s + 4·52-s − 12·59-s + 16·61-s − 8·64-s + 18·69-s − 24·71-s + 4·73-s + 75-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s + 1/3·9-s + 1.80·11-s + 0.577·12-s − 0.554·13-s + 16-s − 3.75·23-s − 1/5·25-s − 0.192·27-s − 1.04·33-s − 1/3·36-s − 1.31·37-s + 0.320·39-s − 1.80·44-s + 1.75·47-s − 0.577·48-s + 2/7·49-s + 0.554·52-s − 1.56·59-s + 2.04·61-s − 64-s + 2.16·69-s − 2.84·71-s + 0.468·73-s + 0.115·75-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124848 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124848 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | $C_1$ | \( 1 + T \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 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
−9.392750473138958647961697264371, −8.610578703883184504620815355064, −8.382258919016960888464931958518, −7.66773143025981285351749668682, −7.25132554126425354900557687617, −6.56329579539546958275321552423, −5.94155321226588727412442600100, −5.81241206259765754250513501442, −5.01602847451418763245834802500, −4.19114262883338708198809215469, −4.10168552133706270535583651662, −3.52343415212056409865262439727, −2.22032145426687050578358306238, −1.39440035412653237109928920217, 0,
1.39440035412653237109928920217, 2.22032145426687050578358306238, 3.52343415212056409865262439727, 4.10168552133706270535583651662, 4.19114262883338708198809215469, 5.01602847451418763245834802500, 5.81241206259765754250513501442, 5.94155321226588727412442600100, 6.56329579539546958275321552423, 7.25132554126425354900557687617, 7.66773143025981285351749668682, 8.382258919016960888464931958518, 8.610578703883184504620815355064, 9.392750473138958647961697264371
