| L(s) = 1 | + 5·9-s + 6·11-s + 10·19-s − 4·31-s − 6·41-s + 10·49-s + 4·61-s − 24·71-s − 20·79-s + 16·81-s − 30·89-s + 30·99-s − 36·101-s + 20·109-s + 5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 50·171-s + 173-s + ⋯ |
| L(s) = 1 | + 5/3·9-s + 1.80·11-s + 2.29·19-s − 0.718·31-s − 0.937·41-s + 10/7·49-s + 0.512·61-s − 2.84·71-s − 2.25·79-s + 16/9·81-s − 3.17·89-s + 3.01·99-s − 3.58·101-s + 1.91·109-s + 5/11·121-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 + 3.82·171-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.288719308\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.288719308\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.46822605447819669916355779686, −11.30153064857169753498901739441, −10.27471433029669895680582697121, −10.24687224730373125705500531262, −9.614882365165757193709607156729, −9.360458524719688217977702554542, −8.870520481978145873329399810057, −8.373520931184517709251016124350, −7.49226383067950484691093404251, −7.32679350906732198766645922100, −6.90983635137271417369129076965, −6.46520486993082766415962116666, −5.58885714756729336601645393372, −5.38090456588438356259620565919, −4.28851712301773069400353975929, −4.24300708169927701793027072627, −3.51634191517170254505242788255, −2.83245529245575853991843794255, −1.48150425037925129244743042230, −1.32083301440450017402421696653,
1.32083301440450017402421696653, 1.48150425037925129244743042230, 2.83245529245575853991843794255, 3.51634191517170254505242788255, 4.24300708169927701793027072627, 4.28851712301773069400353975929, 5.38090456588438356259620565919, 5.58885714756729336601645393372, 6.46520486993082766415962116666, 6.90983635137271417369129076965, 7.32679350906732198766645922100, 7.49226383067950484691093404251, 8.373520931184517709251016124350, 8.870520481978145873329399810057, 9.360458524719688217977702554542, 9.614882365165757193709607156729, 10.24687224730373125705500531262, 10.27471433029669895680582697121, 11.30153064857169753498901739441, 11.46822605447819669916355779686
