| L(s) = 1 | − 2·2-s + 2·4-s + 2·5-s − 4·10-s + 10·13-s − 4·16-s − 6·17-s + 4·20-s − 25-s − 20·26-s + 8·32-s + 12·34-s + 10·37-s + 20·41-s + 2·50-s + 20·52-s − 18·53-s − 24·61-s − 8·64-s + 20·65-s − 12·68-s − 10·73-s − 20·74-s − 8·80-s − 40·82-s − 12·85-s − 10·97-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 0.894·5-s − 1.26·10-s + 2.77·13-s − 16-s − 1.45·17-s + 0.894·20-s − 1/5·25-s − 3.92·26-s + 1.41·32-s + 2.05·34-s + 1.64·37-s + 3.12·41-s + 0.282·50-s + 2.77·52-s − 2.47·53-s − 3.07·61-s − 64-s + 2.48·65-s − 1.45·68-s − 1.17·73-s − 2.32·74-s − 0.894·80-s − 4.41·82-s − 1.30·85-s − 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7914164094\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7914164094\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−13.28805447705671232113784380851, −12.53586617313810056259335698502, −11.41149446976614744378539998100, −11.23982122252934138297467481386, −10.87792335298834723839209377984, −10.50438066563289087637408952687, −9.733438715358766573245468454021, −9.317463580516028159955219232479, −8.897159471699952949424650508928, −8.650779552603517603914584405882, −7.74049675756344181815633037748, −7.64530230168879810686107964962, −6.50151887304805599112061858185, −6.05281541199805411485597576184, −6.03101414069381520735005196939, −4.59652424833388629351632525704, −4.16612828926509674736220026343, −3.02669365898786424020353217412, −1.99879447334211626090229653412, −1.17790720201457529834445921806,
1.17790720201457529834445921806, 1.99879447334211626090229653412, 3.02669365898786424020353217412, 4.16612828926509674736220026343, 4.59652424833388629351632525704, 6.03101414069381520735005196939, 6.05281541199805411485597576184, 6.50151887304805599112061858185, 7.64530230168879810686107964962, 7.74049675756344181815633037748, 8.650779552603517603914584405882, 8.897159471699952949424650508928, 9.317463580516028159955219232479, 9.733438715358766573245468454021, 10.50438066563289087637408952687, 10.87792335298834723839209377984, 11.23982122252934138297467481386, 11.41149446976614744378539998100, 12.53586617313810056259335698502, 13.28805447705671232113784380851
