| L(s) = 1 | + 2-s + 4-s + 8-s − 6·9-s + 16-s − 6·18-s + 2·25-s + 12·29-s + 32-s − 6·36-s + 12·37-s − 7·49-s + 2·50-s + 12·53-s + 12·58-s + 64-s − 6·72-s + 12·74-s + 27·81-s − 7·98-s + 2·100-s + 12·106-s + 28·109-s − 12·113-s + 12·116-s + 121-s + 127-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 2·9-s + 1/4·16-s − 1.41·18-s + 2/5·25-s + 2.22·29-s + 0.176·32-s − 36-s + 1.97·37-s − 49-s + 0.282·50-s + 1.64·53-s + 1.57·58-s + 1/8·64-s − 0.707·72-s + 1.39·74-s + 3·81-s − 0.707·98-s + 1/5·100-s + 1.16·106-s + 2.68·109-s − 1.12·113-s + 1.11·116-s + 1/11·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189728 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189728 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.210217144\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.210217144\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 158 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
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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
−8.997291626899887384834143254771, −8.522246390101857531471954059736, −8.274641504568037282484115409443, −7.75982184044138210927564408635, −7.09150215403131244147991849894, −6.54531944321006279560610218323, −5.98733213940681576713921339220, −5.85996965103172325339290708243, −5.00929014774510108829753339263, −4.74664631908576468087518057764, −3.98206263095044002418964687092, −3.20691175465193001901614030060, −2.79044026532772992918458360057, −2.26587343690739134274856106498, −0.856901038950492236185677643703,
0.856901038950492236185677643703, 2.26587343690739134274856106498, 2.79044026532772992918458360057, 3.20691175465193001901614030060, 3.98206263095044002418964687092, 4.74664631908576468087518057764, 5.00929014774510108829753339263, 5.85996965103172325339290708243, 5.98733213940681576713921339220, 6.54531944321006279560610218323, 7.09150215403131244147991849894, 7.75982184044138210927564408635, 8.274641504568037282484115409443, 8.522246390101857531471954059736, 8.997291626899887384834143254771
