| L(s) = 1 | + 2-s − 4-s − 3·8-s + 4·11-s − 16-s + 8·19-s + 4·22-s + 2·25-s + 5·32-s + 8·38-s − 8·41-s − 12·43-s − 4·44-s + 10·49-s + 2·50-s + 7·64-s + 4·67-s + 12·73-s − 8·76-s − 8·82-s + 12·83-s − 12·86-s − 12·88-s + 4·89-s + 10·98-s − 2·100-s + 16·113-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.06·8-s + 1.20·11-s − 1/4·16-s + 1.83·19-s + 0.852·22-s + 2/5·25-s + 0.883·32-s + 1.29·38-s − 1.24·41-s − 1.82·43-s − 0.603·44-s + 10/7·49-s + 0.282·50-s + 7/8·64-s + 0.488·67-s + 1.40·73-s − 0.917·76-s − 0.883·82-s + 1.31·83-s − 1.29·86-s − 1.27·88-s + 0.423·89-s + 1.01·98-s − 1/5·100-s + 1.50·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.799098161\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.799098161\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 3 | | \( 1 \) |
| 19 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−7.83434718203757685255834046079, −7.22373553724877690748003654670, −6.89982577879480572978497253053, −6.37433087595197550913854169100, −6.15908915766846271301429364994, −5.41598663295276330634734694349, −5.08400571098084286145866880315, −4.95465382946846123568855283953, −4.08796639835778773879693366538, −3.85378827576869401944511594133, −3.29606531706330076680897397861, −3.01072300024435310401946410676, −2.13930686294848903083894902068, −1.36684165213619452723641359730, −0.67473495841023108758814462264,
0.67473495841023108758814462264, 1.36684165213619452723641359730, 2.13930686294848903083894902068, 3.01072300024435310401946410676, 3.29606531706330076680897397861, 3.85378827576869401944511594133, 4.08796639835778773879693366538, 4.95465382946846123568855283953, 5.08400571098084286145866880315, 5.41598663295276330634734694349, 6.15908915766846271301429364994, 6.37433087595197550913854169100, 6.89982577879480572978497253053, 7.22373553724877690748003654670, 7.83434718203757685255834046079
