| L(s) = 1 | − 3·3-s + 2·5-s + 6·9-s − 4·11-s + 3·13-s − 6·15-s − 14·17-s − 10·19-s − 4·23-s + 5·25-s − 9·27-s + 29-s − 3·31-s + 12·33-s + 22·37-s − 9·39-s − 9·41-s − 5·43-s + 12·45-s + 3·47-s + 42·51-s + 6·53-s − 8·55-s + 30·57-s − 7·59-s + 3·61-s + 6·65-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.894·5-s + 2·9-s − 1.20·11-s + 0.832·13-s − 1.54·15-s − 3.39·17-s − 2.29·19-s − 0.834·23-s + 25-s − 1.73·27-s + 0.185·29-s − 0.538·31-s + 2.08·33-s + 3.61·37-s − 1.44·39-s − 1.40·41-s − 0.762·43-s + 1.78·45-s + 0.437·47-s + 5.88·51-s + 0.824·53-s − 1.07·55-s + 3.97·57-s − 0.911·59-s + 0.384·61-s + 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 3 T - 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - T - 28 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 3 T - 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 7 T - 10 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 3 T - 52 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 13 T + 102 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 9 T + 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - T - 82 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 17 T + 192 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
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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
−8.964419060660366002131234864032, −8.789897191943840195728656688343, −8.401828442189882577144039273305, −8.016936254678466207074755604218, −7.12685167474795544954305853700, −7.08375758753444117725567138885, −6.50298383934735771135948085540, −6.09425771595818714518119591384, −6.04788214020341261428847963889, −5.71560435912110414299267860097, −4.78864368759123323322049562980, −4.74107154704612614656140232953, −4.24015753110969675349307242265, −4.06731067756009020546767167620, −2.82680109581761508252724427473, −2.41390891911840451304352225335, −1.97221232381004059382185293125, −1.31596483718426853818580891771, 0, 0,
1.31596483718426853818580891771, 1.97221232381004059382185293125, 2.41390891911840451304352225335, 2.82680109581761508252724427473, 4.06731067756009020546767167620, 4.24015753110969675349307242265, 4.74107154704612614656140232953, 4.78864368759123323322049562980, 5.71560435912110414299267860097, 6.04788214020341261428847963889, 6.09425771595818714518119591384, 6.50298383934735771135948085540, 7.08375758753444117725567138885, 7.12685167474795544954305853700, 8.016936254678466207074755604218, 8.401828442189882577144039273305, 8.789897191943840195728656688343, 8.964419060660366002131234864032
