| L(s) = 1 | − 9-s + 4·11-s + 8·19-s + 4·29-s − 49-s + 16·59-s + 12·61-s + 28·71-s − 24·79-s + 81-s − 4·99-s + 32·101-s + 4·109-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s − 8·171-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 1.20·11-s + 1.83·19-s + 0.742·29-s − 1/7·49-s + 2.08·59-s + 1.53·61-s + 3.32·71-s − 2.70·79-s + 1/9·81-s − 0.402·99-s + 3.18·101-s + 0.383·109-s − 0.909·121-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 − 0.611·171-s + 0.0760·173-s + 0.0747·179-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
| \(L(1)\) |
\(\approx\) |
\(3.091485022\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.091485022\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−9.410842450543863437174383112549, −8.718276897808999994117697582815, −8.660605296538704575010713847983, −8.259656528500359648891736466763, −7.63610894803344717515245711160, −7.42694250125682410306923920463, −6.85739970011255391754245817532, −6.66879814880379010046535380851, −6.19616678962807287090231847464, −5.69515295677129392827215795130, −5.28631737703050010564251201895, −5.01307876736874911345931860191, −4.40875461229187015404561513212, −3.85556634225837624608114609035, −3.57901358921193336593023507932, −3.07819373238062514447643808718, −2.49116600547936501545887035581, −1.94162255760133997711134344667, −1.13047142605742605658667390980, −0.74047661061743712882455587303,
0.74047661061743712882455587303, 1.13047142605742605658667390980, 1.94162255760133997711134344667, 2.49116600547936501545887035581, 3.07819373238062514447643808718, 3.57901358921193336593023507932, 3.85556634225837624608114609035, 4.40875461229187015404561513212, 5.01307876736874911345931860191, 5.28631737703050010564251201895, 5.69515295677129392827215795130, 6.19616678962807287090231847464, 6.66879814880379010046535380851, 6.85739970011255391754245817532, 7.42694250125682410306923920463, 7.63610894803344717515245711160, 8.259656528500359648891736466763, 8.660605296538704575010713847983, 8.718276897808999994117697582815, 9.410842450543863437174383112549
