| L(s) = 1 | − 9-s + 12·11-s − 4·19-s − 12·29-s − 20·31-s − 12·41-s − 49-s + 28·61-s + 12·71-s + 32·79-s + 81-s − 12·89-s − 12·99-s + 12·101-s − 4·109-s + 86·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 4·171-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 3.61·11-s − 0.917·19-s − 2.22·29-s − 3.59·31-s − 1.87·41-s − 1/7·49-s + 3.58·61-s + 1.42·71-s + 3.60·79-s + 1/9·81-s − 1.27·89-s − 1.20·99-s + 1.19·101-s − 0.383·109-s + 7.81·121-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 + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + 0.305·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.329139561\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.329139561\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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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
−9.155355237928304641317401512772, −9.056905264667100048224988577378, −8.646693035088994163307033674817, −8.324123597042410467535872598423, −7.64120898165077936271498657367, −7.29533866146785588398474346426, −6.72968007592384811071474199259, −6.70070818044907245779431159408, −6.34479701767229900640850277198, −5.65456109008119805397148376645, −5.42627908550052787580229016327, −4.97811158927648025340578562254, −4.13010747690806675939735742786, −3.85423576287053218733809732813, −3.63597240633513617544941809826, −3.42843394096223263262900199321, −2.05370804337113063915514180621, −1.99769552241843360414095510257, −1.45130304828502986774155297989, −0.54789257247745583718774084444,
0.54789257247745583718774084444, 1.45130304828502986774155297989, 1.99769552241843360414095510257, 2.05370804337113063915514180621, 3.42843394096223263262900199321, 3.63597240633513617544941809826, 3.85423576287053218733809732813, 4.13010747690806675939735742786, 4.97811158927648025340578562254, 5.42627908550052787580229016327, 5.65456109008119805397148376645, 6.34479701767229900640850277198, 6.70070818044907245779431159408, 6.72968007592384811071474199259, 7.29533866146785588398474346426, 7.64120898165077936271498657367, 8.324123597042410467535872598423, 8.646693035088994163307033674817, 9.056905264667100048224988577378, 9.155355237928304641317401512772
