| L(s) = 1 | + 2·9-s + 10·11-s + 2·19-s + 4·29-s + 8·31-s + 20·41-s + 5·49-s − 12·59-s − 26·61-s + 4·71-s − 16·79-s − 5·81-s − 24·89-s + 20·99-s − 20·101-s + 53·121-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 | + 2/3·9-s + 3.01·11-s + 0.458·19-s + 0.742·29-s + 1.43·31-s + 3.12·41-s + 5/7·49-s − 1.56·59-s − 3.32·61-s + 0.474·71-s − 1.80·79-s − 5/9·81-s − 2.54·89-s + 2.01·99-s − 1.99·101-s + 4.81·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3610000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3610000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.785217514\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.785217514\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 93 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−9.367667422528712285045736977776, −9.006285706843200558185466981691, −8.937247060349751862163973451090, −8.091276856593093183676368912344, −7.989624844374159133301078499952, −7.25242625977625015876359993936, −7.12454474096546215315220123562, −6.56075709709534587264450054775, −6.34403778204241150411354194365, −5.88072618684092498975607396163, −5.60851052840729033448278105850, −4.58142533147572122792800530726, −4.44090391642481129106412961464, −4.17906879501390218202034610650, −3.73642509852256930354574962852, −2.85790354705143152843524253306, −2.82566767330689122621370837526, −1.57369760318378969468882391610, −1.42908619481175922851742105348, −0.815035649131947362621430472197,
0.815035649131947362621430472197, 1.42908619481175922851742105348, 1.57369760318378969468882391610, 2.82566767330689122621370837526, 2.85790354705143152843524253306, 3.73642509852256930354574962852, 4.17906879501390218202034610650, 4.44090391642481129106412961464, 4.58142533147572122792800530726, 5.60851052840729033448278105850, 5.88072618684092498975607396163, 6.34403778204241150411354194365, 6.56075709709534587264450054775, 7.12454474096546215315220123562, 7.25242625977625015876359993936, 7.989624844374159133301078499952, 8.091276856593093183676368912344, 8.937247060349751862163973451090, 9.006285706843200558185466981691, 9.367667422528712285045736977776
