| L(s) = 1 | − 3·9-s − 10·13-s − 20·37-s + 11·49-s − 26·61-s + 20·73-s + 9·81-s − 10·97-s + 38·109-s + 30·117-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 49·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | − 9-s − 2.77·13-s − 3.28·37-s + 11/7·49-s − 3.32·61-s + 2.34·73-s + 81-s − 1.01·97-s + 3.63·109-s + 2.77·117-s − 2·121-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 + 3.76·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4741954575\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4741954575\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
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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
−9.988602571823711885986141381344, −9.514800567329425139879780081743, −9.043489449674298700228258812157, −8.919020044526602431846239970269, −8.305282279399175030971269834257, −7.88082001191237553437677673844, −7.41569766591813217294246981853, −7.14631336816116691728203265300, −6.78317395242220721289247436099, −6.13240513287295041899967070526, −5.70991996374891561132490399374, −5.16068530292061109012537724793, −4.90037123011317828003612267126, −4.57987201727591945190672039310, −3.63911607407508923724077599113, −3.36459917526947416790204583042, −2.52121662760051160279188028485, −2.39643796912090144476165201634, −1.58955996452107888189024399667, −0.27993875093401309848081327225,
0.27993875093401309848081327225, 1.58955996452107888189024399667, 2.39643796912090144476165201634, 2.52121662760051160279188028485, 3.36459917526947416790204583042, 3.63911607407508923724077599113, 4.57987201727591945190672039310, 4.90037123011317828003612267126, 5.16068530292061109012537724793, 5.70991996374891561132490399374, 6.13240513287295041899967070526, 6.78317395242220721289247436099, 7.14631336816116691728203265300, 7.41569766591813217294246981853, 7.88082001191237553437677673844, 8.305282279399175030971269834257, 8.919020044526602431846239970269, 9.043489449674298700228258812157, 9.514800567329425139879780081743, 9.988602571823711885986141381344
