| L(s) = 1 | + 5·9-s − 2·11-s − 2·19-s + 12·29-s + 8·31-s − 10·41-s − 49-s + 8·59-s + 12·61-s + 28·71-s − 28·79-s + 16·81-s + 6·89-s − 10·99-s + 20·109-s − 19·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s − 10·171-s + ⋯ |
| L(s) = 1 | + 5/3·9-s − 0.603·11-s − 0.458·19-s + 2.22·29-s + 1.43·31-s − 1.56·41-s − 1/7·49-s + 1.04·59-s + 1.53·61-s + 3.32·71-s − 3.15·79-s + 16/9·81-s + 0.635·89-s − 1.00·99-s + 1.91·109-s − 1.72·121-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 − 0.764·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.673767385\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.673767385\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 109 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 79 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 165 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 158 T^{2} + p^{2} T^{4} \) |
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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
−9.745358688199461289818865521463, −9.721398022295779136169620725401, −8.700113316836445840078054960900, −8.646280040694369929067538312951, −8.044388465311646721934293537643, −7.981934832072708442320858249336, −7.05525234841462589816395295072, −7.04401845026294622597662082657, −6.60169324507038376327337793672, −6.20906592445487140484728905006, −5.58025139286356655472514970414, −4.98777396072363524285581432063, −4.72547677431999923314001115728, −4.34382941248300225089475117596, −3.76745244395628661583181959798, −3.27803713921887626994531108069, −2.55468970728621955049122464163, −2.15448702032286839235693112090, −1.33234153155859932189094019486, −0.73397297778883513839840362463,
0.73397297778883513839840362463, 1.33234153155859932189094019486, 2.15448702032286839235693112090, 2.55468970728621955049122464163, 3.27803713921887626994531108069, 3.76745244395628661583181959798, 4.34382941248300225089475117596, 4.72547677431999923314001115728, 4.98777396072363524285581432063, 5.58025139286356655472514970414, 6.20906592445487140484728905006, 6.60169324507038376327337793672, 7.04401845026294622597662082657, 7.05525234841462589816395295072, 7.981934832072708442320858249336, 8.044388465311646721934293537643, 8.646280040694369929067538312951, 8.700113316836445840078054960900, 9.721398022295779136169620725401, 9.745358688199461289818865521463
