| L(s) = 1 | + 2·3-s + 4-s + 3·9-s − 11-s + 2·12-s + 16-s − 8·23-s − 10·25-s + 4·27-s − 4·31-s − 2·33-s + 3·36-s + 20·37-s − 44-s − 12·47-s + 2·48-s + 49-s − 20·53-s + 64-s − 8·67-s − 16·69-s + 32·71-s − 20·75-s + 5·81-s − 12·89-s − 8·92-s − 8·93-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/2·4-s + 9-s − 0.301·11-s + 0.577·12-s + 1/4·16-s − 1.66·23-s − 2·25-s + 0.769·27-s − 0.718·31-s − 0.348·33-s + 1/2·36-s + 3.28·37-s − 0.150·44-s − 1.75·47-s + 0.288·48-s + 1/7·49-s − 2.74·53-s + 1/8·64-s − 0.977·67-s − 1.92·69-s + 3.79·71-s − 2.30·75-s + 5/9·81-s − 1.27·89-s − 0.834·92-s − 0.829·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2347884 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2347884 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−7.79089756844307556875120056782, −7.15083214327735671821742085196, −6.50358738357793944503631024403, −6.42153264848366358553087579074, −5.64719531132362490687226713521, −5.61053425475447892671224791397, −4.63059859799734414629334650532, −4.33149302139153054926791784615, −3.88277706681863956744577131696, −3.35278591108779725059175891001, −2.90909027996993607554406630510, −2.22064810010506599000527056391, −2.00017416871630464571063850965, −1.28532701621031142017445445957, 0,
1.28532701621031142017445445957, 2.00017416871630464571063850965, 2.22064810010506599000527056391, 2.90909027996993607554406630510, 3.35278591108779725059175891001, 3.88277706681863956744577131696, 4.33149302139153054926791784615, 4.63059859799734414629334650532, 5.61053425475447892671224791397, 5.64719531132362490687226713521, 6.42153264848366358553087579074, 6.50358738357793944503631024403, 7.15083214327735671821742085196, 7.79089756844307556875120056782
