| L(s) = 1 | − 2·7-s + 2·9-s + 4·17-s − 6·25-s + 16·31-s + 20·41-s − 16·47-s + 3·49-s − 4·63-s − 16·71-s − 12·73-s − 16·79-s − 5·81-s + 20·89-s + 4·97-s + 32·103-s + 4·113-s − 8·119-s + 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 8·153-s + 157-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 2/3·9-s + 0.970·17-s − 6/5·25-s + 2.87·31-s + 3.12·41-s − 2.33·47-s + 3/7·49-s − 0.503·63-s − 1.89·71-s − 1.40·73-s − 1.80·79-s − 5/9·81-s + 2.11·89-s + 0.406·97-s + 3.15·103-s + 0.376·113-s − 0.733·119-s + 1.63·121-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 + 0.646·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.635738767\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.635738767\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−11.46956331397052785848454212106, −10.79551282548855389627948135389, −10.14845594714283680709060130124, −10.00078034210602862699336737032, −9.785440064046677561822638182324, −9.078664817954983034676137389722, −8.697769018473295927210536655281, −8.033104611333762006890392439678, −7.52069612891295355221778290769, −7.44222462391731258193780523312, −6.47455121033329918206541337099, −6.22330473762396035709990404524, −5.84301046069638269995488748734, −5.07135260317979897949845342147, −4.33115154610660680330821545485, −4.17607887309210232108380887483, −3.12865799059035804272392854294, −2.88507441518943687830078845100, −1.83048307998533227420738297881, −0.848289605798021387670017183136,
0.848289605798021387670017183136, 1.83048307998533227420738297881, 2.88507441518943687830078845100, 3.12865799059035804272392854294, 4.17607887309210232108380887483, 4.33115154610660680330821545485, 5.07135260317979897949845342147, 5.84301046069638269995488748734, 6.22330473762396035709990404524, 6.47455121033329918206541337099, 7.44222462391731258193780523312, 7.52069612891295355221778290769, 8.033104611333762006890392439678, 8.697769018473295927210536655281, 9.078664817954983034676137389722, 9.785440064046677561822638182324, 10.00078034210602862699336737032, 10.14845594714283680709060130124, 10.79551282548855389627948135389, 11.46956331397052785848454212106
