L(s) = 1 | − 2·4-s + 9-s + 4·16-s − 4·31-s − 2·36-s − 12·41-s − 2·49-s − 8·64-s − 24·71-s + 20·79-s + 81-s − 12·89-s + 10·121-s + 8·124-s + 127-s + 131-s + 137-s + 139-s + 4·144-s + 149-s + 151-s + 157-s + 163-s + 24·164-s + 167-s − 22·169-s + 173-s + ⋯ |
L(s) = 1 | − 4-s + 1/3·9-s + 16-s − 0.718·31-s − 1/3·36-s − 1.87·41-s − 2/7·49-s − 64-s − 2.84·71-s + 2.25·79-s + 1/9·81-s − 1.27·89-s + 0.909·121-s + 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1/3·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 1.87·164-s + 0.0773·167-s − 1.69·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | | \( 1 \) |
good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 146 T^{2} + p^{2} T^{4} \) |
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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
−8.471735651756358886191812992807, −8.136073050603030780287194942213, −7.62737050745987647003109275481, −7.10939010094640721565819959302, −6.69996816528396882333362272571, −5.99032722863745092835858749406, −5.63401665125598587390415662685, −4.97363524286420755586596678674, −4.68582914817178728463656973298, −4.03692737540833527234822179494, −3.52093714862999742208675072166, −3.00194017979805245569288086992, −2.00682666441558622563445980096, −1.24174137710428244225262808466, 0,
1.24174137710428244225262808466, 2.00682666441558622563445980096, 3.00194017979805245569288086992, 3.52093714862999742208675072166, 4.03692737540833527234822179494, 4.68582914817178728463656973298, 4.97363524286420755586596678674, 5.63401665125598587390415662685, 5.99032722863745092835858749406, 6.69996816528396882333362272571, 7.10939010094640721565819959302, 7.62737050745987647003109275481, 8.136073050603030780287194942213, 8.471735651756358886191812992807