| L(s) = 1 | − 2·3-s + 2·5-s + 3·9-s − 4·15-s + 8·19-s + 3·25-s − 4·27-s + 4·31-s + 6·45-s + 14·49-s − 16·57-s + 12·59-s + 20·61-s − 8·67-s − 28·73-s − 6·75-s − 28·79-s + 5·81-s − 8·93-s + 16·95-s − 12·101-s + 8·103-s + 24·107-s + 10·121-s + 4·125-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s + 9-s − 1.03·15-s + 1.83·19-s + 3/5·25-s − 0.769·27-s + 0.718·31-s + 0.894·45-s + 2·49-s − 2.11·57-s + 1.56·59-s + 2.56·61-s − 0.977·67-s − 3.27·73-s − 0.692·75-s − 3.15·79-s + 5/9·81-s − 0.829·93-s + 1.64·95-s − 1.19·101-s + 0.788·103-s + 2.32·107-s + 0.909·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20793600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20793600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.849554036\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.849554036\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
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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
−8.437775599658664199421713722645, −8.341147327758749115538787649213, −7.54125323313774272517193289300, −7.24809641918808843436232034512, −6.96463524476219341932812577747, −6.92534094657705784984949006646, −6.10482747075722118112335732785, −5.78452088221382407678548648991, −5.58473993323060728155509761557, −5.55881970454292824947434639817, −4.68750815163490212377674520441, −4.65715494166567413702172273426, −4.11825962885721029810828318202, −3.61580191497167478409409978411, −2.90046453652610337549017634835, −2.86205684627799366378172938684, −1.98793297988875652447190692327, −1.65642787595182014946189242962, −0.900495170910319867699830737912, −0.65940472143622125388689901843,
0.65940472143622125388689901843, 0.900495170910319867699830737912, 1.65642787595182014946189242962, 1.98793297988875652447190692327, 2.86205684627799366378172938684, 2.90046453652610337549017634835, 3.61580191497167478409409978411, 4.11825962885721029810828318202, 4.65715494166567413702172273426, 4.68750815163490212377674520441, 5.55881970454292824947434639817, 5.58473993323060728155509761557, 5.78452088221382407678548648991, 6.10482747075722118112335732785, 6.92534094657705784984949006646, 6.96463524476219341932812577747, 7.24809641918808843436232034512, 7.54125323313774272517193289300, 8.341147327758749115538787649213, 8.437775599658664199421713722645
