| L(s) = 1 | + 2-s + 3-s − 4-s + 6-s − 3·8-s + 9-s − 3·11-s − 12-s + 4·13-s − 16-s + 18-s − 3·22-s − 8·23-s − 3·24-s − 6·25-s + 4·26-s + 27-s + 5·32-s − 3·33-s − 36-s − 4·37-s + 4·39-s + 3·44-s − 8·46-s + 8·47-s − 48-s + 2·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s − 1.06·8-s + 1/3·9-s − 0.904·11-s − 0.288·12-s + 1.10·13-s − 1/4·16-s + 0.235·18-s − 0.639·22-s − 1.66·23-s − 0.612·24-s − 6/5·25-s + 0.784·26-s + 0.192·27-s + 0.883·32-s − 0.522·33-s − 1/6·36-s − 0.657·37-s + 0.640·39-s + 0.452·44-s − 1.17·46-s + 1.16·47-s − 0.144·48-s + 2/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4752 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4752 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.115459211\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.115459211\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−12.40034919641662132775971763573, −11.94079317736327500240864866855, −11.24926867346879304788630911023, −10.43264856980990269535257796694, −9.975026179182169205209018836422, −9.304691211808039522291402854248, −8.601379066356035322593763260494, −8.146840241884356568250815540150, −7.53321652078339631746735295713, −6.43759152388885102986219540336, −5.80118775057511944644355096785, −5.13576111489458385632317052281, −4.03826779880850438115451415591, −3.63561346253081326411157377370, −2.35238971472037862013416227689,
2.35238971472037862013416227689, 3.63561346253081326411157377370, 4.03826779880850438115451415591, 5.13576111489458385632317052281, 5.80118775057511944644355096785, 6.43759152388885102986219540336, 7.53321652078339631746735295713, 8.146840241884356568250815540150, 8.601379066356035322593763260494, 9.304691211808039522291402854248, 9.975026179182169205209018836422, 10.43264856980990269535257796694, 11.24926867346879304788630911023, 11.94079317736327500240864866855, 12.40034919641662132775971763573
