| L(s) = 1 | − 3-s + 4-s + 9-s − 12-s + 16-s + 19-s + 25-s − 27-s + 17·29-s + 36-s + 41-s − 9·43-s − 48-s − 13·49-s + 15·53-s − 57-s − 7·59-s − 2·61-s + 64-s + 14·71-s − 8·73-s − 75-s + 76-s + 81-s − 17·87-s − 2·89-s + 100-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/2·4-s + 1/3·9-s − 0.288·12-s + 1/4·16-s + 0.229·19-s + 1/5·25-s − 0.192·27-s + 3.15·29-s + 1/6·36-s + 0.156·41-s − 1.37·43-s − 0.144·48-s − 1.85·49-s + 2.06·53-s − 0.132·57-s − 0.911·59-s − 0.256·61-s + 1/8·64-s + 1.66·71-s − 0.936·73-s − 0.115·75-s + 0.114·76-s + 1/9·81-s − 1.82·87-s − 0.211·89-s + 1/10·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 740772 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740772 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.891491668\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.891491668\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( 1 + T \) |
| 19 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 36 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 39 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 105 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
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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
−8.152025660908239030205934099555, −7.990015172186437913392116207288, −7.25238855970132591132487020796, −6.80786445563556260856664582317, −6.57895209564735994948151257858, −6.13883804438330140993212515652, −5.58081648047654174908596658332, −5.05916911691596608199504186861, −4.64516229073285011960678864689, −4.20975423508657598750437296185, −3.33222945265758899268895072413, −3.00515423082943548256771081130, −2.28705329279368846784334194802, −1.50224245585708132070581986573, −0.73186612623875233427404397957,
0.73186612623875233427404397957, 1.50224245585708132070581986573, 2.28705329279368846784334194802, 3.00515423082943548256771081130, 3.33222945265758899268895072413, 4.20975423508657598750437296185, 4.64516229073285011960678864689, 5.05916911691596608199504186861, 5.58081648047654174908596658332, 6.13883804438330140993212515652, 6.57895209564735994948151257858, 6.80786445563556260856664582317, 7.25238855970132591132487020796, 7.990015172186437913392116207288, 8.152025660908239030205934099555
