| L(s) = 1 | + 7-s + 4·9-s + 2·11-s + 10·23-s − 2·25-s − 4·29-s + 2·43-s + 49-s − 2·53-s + 4·63-s + 24·67-s + 2·77-s + 20·79-s + 7·81-s + 8·99-s − 20·107-s + 4·109-s − 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 10·161-s + 163-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 4/3·9-s + 0.603·11-s + 2.08·23-s − 2/5·25-s − 0.742·29-s + 0.304·43-s + 1/7·49-s − 0.274·53-s + 0.503·63-s + 2.93·67-s + 0.227·77-s + 2.25·79-s + 7/9·81-s + 0.804·99-s − 1.93·107-s + 0.383·109-s − 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.788·161-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.980676135\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.980676135\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 116 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
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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
−7.912223619388828719218453418641, −7.48954510933973185949748831576, −7.02758538217990604983669556002, −6.69795050789738161590665220612, −6.42456172805613540382324189513, −5.67818376926923595677112368581, −5.09753607170920599079661324139, −4.99876694141571338801857518541, −4.26225609717787684250690620531, −3.88131552661872927462766323524, −3.47230161895129920517339350918, −2.70429822496523140810220791599, −2.06653536437247577356060299608, −1.41174998200023901461426944023, −0.835464220794316884090485614816,
0.835464220794316884090485614816, 1.41174998200023901461426944023, 2.06653536437247577356060299608, 2.70429822496523140810220791599, 3.47230161895129920517339350918, 3.88131552661872927462766323524, 4.26225609717787684250690620531, 4.99876694141571338801857518541, 5.09753607170920599079661324139, 5.67818376926923595677112368581, 6.42456172805613540382324189513, 6.69795050789738161590665220612, 7.02758538217990604983669556002, 7.48954510933973185949748831576, 7.912223619388828719218453418641
