| L(s) = 1 | + 2·3-s + 3·9-s + 6·13-s − 12·17-s + 16·23-s − 6·25-s + 4·27-s + 12·29-s + 12·39-s − 8·43-s + 10·49-s − 24·51-s − 20·53-s − 4·61-s + 32·69-s − 12·75-s + 5·81-s + 24·87-s + 12·101-s + 16·103-s − 8·107-s − 12·113-s + 18·117-s + 18·121-s + 127-s − 16·129-s + 131-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s + 1.66·13-s − 2.91·17-s + 3.33·23-s − 6/5·25-s + 0.769·27-s + 2.22·29-s + 1.92·39-s − 1.21·43-s + 10/7·49-s − 3.36·51-s − 2.74·53-s − 0.512·61-s + 3.85·69-s − 1.38·75-s + 5/9·81-s + 2.57·87-s + 1.19·101-s + 1.57·103-s − 0.773·107-s − 1.12·113-s + 1.66·117-s + 1.63·121-s + 0.0887·127-s − 1.40·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.432249822\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.432249822\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
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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
−11.67742979012573206868120495160, −11.43657920178970095993615599365, −10.90079055281521496391493721133, −10.60843076004122407086763511364, −10.09034929326524269194283794390, −9.093025499803326964387221212222, −9.070456123070665916918025519974, −8.851926842741046661304085753495, −8.191062176165377562891932796109, −7.80979920227402879121197521368, −6.92755574353117200056568174982, −6.59823322750470748819114060489, −6.39192484901194424268575474235, −5.31681838809082653166237421121, −4.53540613555075457732054973802, −4.39367577433226799856671426843, −3.38471134054077733832054954836, −2.97713835290684456108862309652, −2.18071165988872254726303044855, −1.23341732032264811311135370976,
1.23341732032264811311135370976, 2.18071165988872254726303044855, 2.97713835290684456108862309652, 3.38471134054077733832054954836, 4.39367577433226799856671426843, 4.53540613555075457732054973802, 5.31681838809082653166237421121, 6.39192484901194424268575474235, 6.59823322750470748819114060489, 6.92755574353117200056568174982, 7.80979920227402879121197521368, 8.191062176165377562891932796109, 8.851926842741046661304085753495, 9.070456123070665916918025519974, 9.093025499803326964387221212222, 10.09034929326524269194283794390, 10.60843076004122407086763511364, 10.90079055281521496391493721133, 11.43657920178970095993615599365, 11.67742979012573206868120495160
