| L(s) = 1 | + 2-s − 3-s − 5-s − 6-s + 7-s − 8-s + 9-s − 10-s + 14-s + 15-s − 16-s + 18-s − 21-s − 23-s + 24-s − 2·27-s − 2·29-s + 30-s − 35-s + 40-s − 2·41-s − 42-s + 2·43-s − 45-s − 46-s + 2·47-s + 48-s + ⋯ |
| L(s) = 1 | + 2-s − 3-s − 5-s − 6-s + 7-s − 8-s + 9-s − 10-s + 14-s + 15-s − 16-s + 18-s − 21-s − 23-s + 24-s − 2·27-s − 2·29-s + 30-s − 35-s + 40-s − 2·41-s − 42-s + 2·43-s − 45-s − 46-s + 2·47-s + 48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4070629205\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4070629205\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−13.72109285787297334621672318409, −12.91953457458712685832370021318, −12.78482575618033582120647928994, −11.89730415182283369245167736241, −11.71208045192582949871815458486, −11.59390338033710325339197060635, −10.67564303383999914924915166349, −10.49131899340443749749830472998, −9.392275456734232582728614569972, −9.186924837552491493319425722300, −8.208014038692492856373607073400, −7.71848071459985778649623621210, −7.29984296400960119785591340565, −6.44825492248817334948738652479, −5.67425283931826133655283997430, −5.42165937945702167287681403977, −4.62317910412855121571655553895, −3.98412937626303021390155283110, −3.69435910234721680945246195208, −2.07304643702320871970159871316,
2.07304643702320871970159871316, 3.69435910234721680945246195208, 3.98412937626303021390155283110, 4.62317910412855121571655553895, 5.42165937945702167287681403977, 5.67425283931826133655283997430, 6.44825492248817334948738652479, 7.29984296400960119785591340565, 7.71848071459985778649623621210, 8.208014038692492856373607073400, 9.186924837552491493319425722300, 9.392275456734232582728614569972, 10.49131899340443749749830472998, 10.67564303383999914924915166349, 11.59390338033710325339197060635, 11.71208045192582949871815458486, 11.89730415182283369245167736241, 12.78482575618033582120647928994, 12.91953457458712685832370021318, 13.72109285787297334621672318409
