L(s) = 1 | − 2·3-s − 2·5-s + 2·7-s + 3·9-s + 4·15-s − 4·21-s + 4·23-s + 3·25-s − 4·27-s − 8·29-s − 4·31-s − 4·35-s − 8·37-s + 4·41-s − 12·43-s − 6·45-s + 4·47-s + 3·49-s − 4·53-s + 8·59-s − 16·61-s + 6·63-s + 4·67-s − 8·69-s + 4·71-s + 8·73-s − 6·75-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 0.894·5-s + 0.755·7-s + 9-s + 1.03·15-s − 0.872·21-s + 0.834·23-s + 3/5·25-s − 0.769·27-s − 1.48·29-s − 0.718·31-s − 0.676·35-s − 1.31·37-s + 0.624·41-s − 1.82·43-s − 0.894·45-s + 0.583·47-s + 3/7·49-s − 0.549·53-s + 1.04·59-s − 2.04·61-s + 0.755·63-s + 0.488·67-s − 0.963·69-s + 0.474·71-s + 0.936·73-s − 0.692·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45158400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45158400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_4$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_4$ | \( 1 + 16 T + 166 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T - 34 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4 T + 102 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 174 T^{2} + p^{2} T^{4} \) |
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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
−7.58938889458375342114894915733, −7.58400705768765675929275648209, −7.04067241544700805494926835356, −6.86185048860045792638341218276, −6.32014825213729218481076959664, −6.19601871327697399192661268510, −5.40197774647185895400154717762, −5.38938401713916874867455528795, −4.90500382662263215337757360449, −4.85546647127553761811008252183, −4.11736709701848401100996505607, −3.92537317304507086089953496209, −3.43444876228649193080650819358, −3.22105032619658078629174650559, −2.25594340468474005481396056674, −2.12506417643249451370148878656, −1.23320469953243744914173468680, −1.16515863644149473107174368034, 0, 0,
1.16515863644149473107174368034, 1.23320469953243744914173468680, 2.12506417643249451370148878656, 2.25594340468474005481396056674, 3.22105032619658078629174650559, 3.43444876228649193080650819358, 3.92537317304507086089953496209, 4.11736709701848401100996505607, 4.85546647127553761811008252183, 4.90500382662263215337757360449, 5.38938401713916874867455528795, 5.40197774647185895400154717762, 6.19601871327697399192661268510, 6.32014825213729218481076959664, 6.86185048860045792638341218276, 7.04067241544700805494926835356, 7.58400705768765675929275648209, 7.58938889458375342114894915733